A mysterious company owned by an unknown foreign country is continuing a battle over an effort by a prosecutor — who quite possibly is special counsel Robert Mueller — to subpoena information from that corporation.

The fight could go all the way to the Supreme Court.

The battle, waged almost completely out of sight from the public, has ignited intense speculation in Washington about what the company does, what country owns it, whether the information sought relates to Mueller's sprawling criminal probe — and if that information relates to President Donald Trump or people close to him.

On Saturday the company asked the U.S .Supreme Court to step in after it lost an appeal in federal court. If the Supreme Court takes up this mystery case, experts say it could be the first time in SCOTUS history that the justices hear a case that's entirely under seal.

Earlier in the week, a panel of three judges on a federal appeals court revealed for the first time that the entity trying to keep its information secret was a company owned by what was identified only as "Country A."

The appeals court said the company must turn over the information sought by prosecutors.

Before that ruling, the identity of the appellant in the case was unknown.

The shadowy fight began in mid-August in the U.S. District Court for the District of Columbia, where a grand jury investigating a criminal matter had issued a subpoena for information on the firm.

Almost as a rule, a grand jury would take such an action at the request of a prosecutor who has been presenting evidence to the member jurors.

The company then filed a legal action seeking to quash the subpoena in the same court. The company's lawsuit appears as "Sealed vs. Sealed" on the federal court system's computer network.

Judge Beryl Howell rejected the company's effort. And Howell also imposed a $5,000 per week fine after she said the company was in contempt for still refusing to comply with the subpoena, records indicate.

Then, on Oct. 10, the company filed a notice of appeal with the federal appeals court in Washington, where the case is known as Number 18-3071, "In re: Grand Jury Subpoena."

The company, appeals records show, was arguing that it was immune from the subpoena under the Foreign Sovereign Immunities Act. The firm also argued that the subpoena was "unreasonable and oppressive" under federal criminal procedure rules because it would require the company to violate the law of the country that owns it.

On Oct. 24, Politico reported that the case may be related to Mueller's probe. The special counsel is investigating Russian interference in the 2016 presidential election, possible collusion with Russians by Trump's campaign and possible obstruction of justice by the president. Trump has denied any wrongdoing.

Politico said that one of its reporters earlier in October had been inside the appeals clerk's office on a day when a filing in the subpoena-related appeals case was due, and overheard a man ask "for a copy of the special counsel's latest sealed filing so that the man's law firm could craft its response." The man declined to identify himself or the client.

"Three hours later, a sealed response in the grand-jury dispute was submitted to the D.C. Circuit," Politico wrote.

CNN reported that its own reporters in early September saw several prosecutors from Mueller's office enter Howell's courtroom not long after the judge issued a ruling upholding the grand jury subpoena. The same prosecutors were spotted entering Howell's courtroom on Oct. 5, after the appeals court referred the case back to her for a hearing.

Mueller's spokesman, Peter Carr, told CNBC on Friday that he could not comment on whether the special counsel was involved in the case.

The appeals court on Dec. 14 heard oral arguments on the company's quash request. Before the arguments began, the entire floor containing the courtroom where the hearing was held was closed to the public.

On Tuesday, the three-judge panel that heard the case issued its ruling, rejecting each of the claims raised by the company.

On Thursday, the company filed a motion that is sealed from public view. The motion is nearly 4,000 words long, according to an entry on the appeals court website.

Two hours after that motion was filed, a judicial order was issued in the case. That order likewise is sealed.

Entities that lose an appeal heard by a panel of three judges can ask for a so-called en banc review of the case.

If that request is granted, a panel comprised of all of the judges of the appeals court would consider the arguments.

If the foreign company in this case is not granted en banc status, or if it loses at that stage, it could ask the U.S. Supreme Court to consider its case.

However, the high court is not required to take the case. And if the Supreme Court rejects the request, the appeals court's decision would stand.