If I'm reading it correctly, the SCOTUS ruled the House and Senate can do pretty much whatever they deem necessary to end obstructions:
https://supreme.justia.com/cases/federal/us/294/125/
Syllabus
1. The power of a House of Congress to punish a private citizen who obstructs the performance of its legislative duties is not limited to the removal of an existing obstruction, but continues after the obstruction has ceased or its removal has become impossible. P. 294 U. S. 147.
Held in this case that the Senate had power to cite for contempt a witness charged with having permitted the removal and destruction of papers which he had been subpoenaed to produce.
2. The Act making refusal to answer or to produce papers before either House, or one of its committees, a misdemeanor (R.S. § 102) did not impair, but supplemented, the power of the House affected to punish for such contempt. P. 294 U. S. 151.
3. Punishment, purely as such, through contempt proceedings, legislative or judicial, is not precluded because punishment may also be inflicted for the same act as a statutory offence. P. 294 U. S. 151.
4. Where a proceeding for contempt is within the jurisdiction of a House of Congress, the questions whether the person arrested is guilty or has so far purged himself that he does not deserve punishment are questions for that House to decide, and which cannot be inquired into by a court by a writ of habeas corpus. P. 294 U. S. 152.
63 App.D.C. 342 72 F. 2d 560, reversed.
Supreme Court, D.C., affirmed.
Certiorari, 293 U.S. 543, to review the reversal of a judgment discharging a writ of habeas corpus by which the above-named respondent sought to gain his release from the custody of the above-named petitioner, the Sergeant-at-Arms of the Senate.
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