After more than two-thirds of the Senate cheered the passage of an immigration bill last week, a bipartisan duo laid down an important milestone in an even heavier legislative lift: tax reform.

Sen. Max Baucus (D) of Montana and Sen. Orrin Hatch (R) of Utah came to the floor only a few minutes after passing the immigration reform bill – a top priority for leaders in both parties – to tell senators it was time, in essence, to eat their spinach.

The two men, the chairman and top Republican on the Senate Finance Committee, will build their overhaul of the nation’s tax code on what’s known as the “zero plan,” they said, meaning that all tax breaks in the corporate and personal income laws are, as of now, out – unless lawmakers make the case for them to be put back in.

“We have made tremendous progress and our now entering the home stretch,” said Senator Hatch. “We are here today to call on all of our colleagues to provide their input to help get tax reform over the finish line.”

But to get to the finish line, senators are going to have to stick out their necks: If there’s a tax preference they’d like to defend, they have until July 26 to speak up.

“The senator’s ‘blank slate’ approach places the burden on those who are protecting specific preferences in the code and asks them to explain why they should have a lower tax rate than everyone else. The government should no longer pick winners and losers among industries, and it is our hope that this effort will result in a simplified and fairer tax code,” said Bill Hughes, a senior vice president at the Retail Industry Leader’s Association in a statement.

There are some preferences (like the earned income tax credit for the working poor, perhaps), Senator Baucus noted, that would likely be kept to make sure the code does not increase tax benefits for the rich at the expense of lower income Americans.

“The blank slate is, of course, not the end of the discussion – you don’t clear the decks and stop,” Baucus said. “Some of the provisions in the code serve very important objectives.”

But to drive home the point that every tax preference hurts the tax reform cause of lowering rates, the senators offered analysis from the Joint Committee on Taxation showing that every $200 billion in annual tax expenditures senators want to put back into the code means another uptick of a percentage point or two in each of the nation’s current six tax brackets.

In the corporate code, every $20 billion in annual expenditures knocks up the rate by 1.5 percent.

How low could US rates go by devouring most of the tax codes cutouts and exceptions? The Simpson-Bowles Commission, put together by President Obama to evaluate ways to cut the deficit, also used a “zero plan” as the basis for their rewrite of the tax code. By sweeping away nearly all tax credits, the commission was able to collapse the current six tax brackets, which range up to 39.6 percent, into three brackets at 9, 15, and 24 percent.

Sprinkling back only a handful of major tax preferences, such as a form of the mortgage interest deduction and the child tax credit, the commission was able to achieve rates of 12, 20 and 27 percent.

Baucus and Hatch’s effort is in the same spirit, if not the precise form, that Rep. Dave Camp (R) of Michigan is pushing in the House in the Ways and Means Committee.

Both chambers have produced a series of tax reform proposals on different parts of the tax code.

Baucus and Representative Camp have been assiduously courting their colleagues both inside their committees and beyond over the last year. Both have canvassed a wide swath of lawmakers on both sides of the aisle, with Baucus having met with every one of his Senate colleagues on the matter.

And the two chairmen will be taking their tax reform show on the road, touring the country together (beginning with two events in Minnesota next week) to take their tax reform cause beyond Washington.

It’s a level of hustle that shows that, for tax reform, it’s probably now or never. Baucus is retiring after six terms in the Senate in 2014. Camp is term-limited atop the Ways and Means Committee.

Simpson and Bowles called the current tax reform drive the “last, best chance” at achieving a rewrite of the nation’s tax laws, which have been growing like kudzu since the last sweeping changes in 1986.

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(For emphasis, Baucus brought all 18 pounds of the tax code to the Senate floor with him last week in paperback binding. “Just think how heavy it’d be in hard cover!” he said.)

“It took three years to get the 1986 bill done,” Hatch said last week. “I don’t think we have three years. We have to do it now, or else it won’t get done.”