Washington (CNN) Nearly a decade after President Barack Obama signed the legislation, and after it twice survived challenges at the Supreme Court, the Affordable Care Act faces a momentous test in a New Orleans courtroom this week.

Millions of Americans, including those with cancer, diabetes and other chronic conditions, cannot be denied coverage because of the ACA's sweeping insurance regulations. With this fresh case, destined to climax at the Supreme Court yet again, the stakes for the continued existence of the ACA are as high as ever.

The case, to be heard Tuesday by a three-judge appeals court panel, was initiated by Texas and other Republican-led states. Joined now fully by the Department of Justice, they want the ACA declared unconstitutional. At an earlier stage of this litigation, the Trump administration had said only certain parts of the law, tied to the individual insurance requirement, should be struck down.

The drama on this round is intensified by the overarching clash between the Republican administration of Trump, who has tried in vain to win repeal of the ACA , and the now Democratic-led House of Representatives, which has intervened to try to keep the law in the place. The administration wants the judiciary to do what it could not win legislatively: eliminate the entire 2010 law.

Ultimate resolution of the case would affect the cost and quality of health care in America. At the same time, amid rising conflicts among branches of government, the case of Texas v. United States tests lines of power between Congress and the courts.

US District Court Judge Reed O'Connor, whose decision declaring the ACA unconstitutional is before the 5th Circuit, brushed aside arguments of Congress' intentions when it passed the ACA in 2010 and amended it in 2017. He also dismissed the traditional view that judges should respect parts of a challenged law not deemed unconstitutional.

Many of the health-care industry and legal advocates who were ringside at previous ACA disputes, for and against, have joined the current fight with "friend of the court" filings to the New Orleans-based circuit.

The crux of arguments from the Trump administration and from Texas and other Republican-led states that began the case is: When the Supreme Court upheld the individual insurance mandate, the ACA linchpin, in 2012, the justices relied on Congress' taxing power. When Congress in 2017 reduced the tax penalty (for those who lacked insurance) to zero, the individual mandate could no longer be deemed constitutional. And, the challengers conclude, because the individual mandate is intertwined with a multitude of ACA provisions, its invalidity sinks the whole law.

Defenders of the ACA, which include California and other Democratic-run states, as well as the House of Representatives, counter that Congress' 2017 action affected only the amount of the tax penalty. They say that if lawmakers wanted to repeal ACA regulations, they would have done so.

Both sides rely on various lines from the Supreme Court's 2012 and 2015 ACA decisions, as suits their respective purposes. Chief Justice John Roberts cast the decisive vote in 2012 to uphold the law, based on Congress' taxing power, rather than a broader assertion of its authority to regulate interstate commerce. Roberts wrote the controlling opinion in the 2012 case known as National Federal of Independent Business v. Sebelius, as well as in the 2015 King v. Burwell decision that affirmed the ACA.

In both decisions, Roberts said judges should read the words of a statute in context, with an eye toward Congress' intentions, which he described here as to improve health-insurance markets.

"In a democracy," Roberts wrote in 2015, "the power to make the law rests with those chosen by the people. Our role is confined -- to say what the law is."

The Texas challengers highlight, however, Roberts' sentiment that the federal government generally "does not have the power to order people to buy health insurance," and they stress that the "essential feature" of the tax penalty led to Roberts' conclusion that the ACA could be upheld.

California and other states defending the law point to Roberts' assertion that judges must look to "a fair understanding" of Congress' plan. They emphasize the chief justice's statement in King v. Burwell: "(I)n every case we must respect the role of the Legislature, and take care not to undo what it has done."

Getting across the legal threshold

The 5th Circuit judges hearing the case Tuesday are Carolyn Dineen King, appointed by President Jimmy Carter in 1979; Jennifer Walker Elrod, named by President George W. Bush in 2007, and Kurt Engelhardt, appointed by Trump in in 2018. Before that three-judge panel addresses the merits of the dispute, it will consider the legal "standing" of the various parties.

Each side argues that the other should not even be in court, and the 5th Circuit signaled in a request for additional filings in June that it particularly wants to explore the "standing" grounds asserted by the Democratic states and House that have intervened.

To establish standing, a party must show it would be harmed, and in this situation, the respective claims mainly center on financial costs. Texas says that the ACA burdens states by bringing more people into state-administered coverage programs. On the other side, California contends that if the ACA disappears, states would lose hundreds of billions of dollars in federal funds, for example, through the ACA's expansion of Medicaid money funneled to cover poor people.

Lawyers for the US House argue that it has an interest in enforcement of a federal law when the Justice Department (which normally would defend an act of Congress) argues that it is unconstitutional.

Beyond such dry but crucial procedural issues, the dueling parties disagree over how Congress' action zeroing out the tax penalty in 2017 may have transformed the constitutionality of the ACA. (That reduction took affect this year.)

"Bereft of penalties," Texas officials argue in their brief, "the mandate now raises no revenue and therefore cannot by any conceivable definition be considered a tax. Stripped of its tax status, the individual mandate is nothing more than an unconstitutional congressional mandate to purchase health insurance."

Texas adds that the individual mandate "necessarily brings down the rest of the ACA" because it was "essential to the Act's community-rating and guaranteed-issue provisions."

The "community-rating" provision prevents insurance companies from charging people higher premiums because of their health history. The "guaranteed-issue" provision requires insurers to cover people regardless of preexisting conditions such as cancer.

On the other side, California officials and their allies emphasize that when Congress zeroed-out the tax penalty for those who did not obtain insurance, it kept all other ACA provisions in place. As a result, they argue, Judge O'Connor's decision "conflicts with the plain intent of Congress and would create chaos and harm tens of millions of Americans."

Next stop the Supreme Court

A 5th Circuit decision is not expected for months, but when it comes, the losing side is likely to appeal to the Supreme Court. Despite changes among the nine in recent years, the five justices who upheld the ACA remain on the bench: Chief Justice Roberts and Associate Justices Ruth Bader Ginsburg, Stephen Breyer, Sonia Sotomayor and Elena Kagan.

Roberts' views are bound to matter most. Since his 2012 move in favor of the ACA, which was the result of switched positions in private negotiations with justices, his role has only become more crucial. On myriad issues beyond Obamacare, Roberts resides at the ideological middle as well as in the chief's center chair. He appears to be moving more cautiously than ever, yet he defies easy forecast.

In the recently completed annual session, for example, Roberts led the conservative majority in a highly consequential decision shutting the door to any federal lawsuit against extreme partisan gerrymanders. With today's sophisticated data techniques, the political party controlling a statehouse when voting-district lines are drawn can entrench its power.

Yet Roberts also wrote a decision -- with shifting majorities across the ideological divide -- that gave the Commerce Secretary considerable latitude to decide 2020 census questions but found, regarding the proposed citizenship-status query, that the rationale offered was "contrived." The Trump administration is still trying to add a citizenship question to next year's decennial census.

In the new ACA dispute, the role of the judiciary is as much at the fore as are the complexities of health care. And attorneys for the US House hit hard the theme of judicial restraint.

"At bottom," they assert, "[Texas] plaintiffs and DOJ invite this Court to affirm a massive judicial overreach that would deny vitally important care to millions, sow chaos in healthcare markets throughout the country, and contravene fundamental principles that should guide interpretation of an Act of Congress."