As popular a rallying cry as "Inherent Contempt!" is on DU, it is highly unlikely that invoking it would ever result in anyone being "locked up."
First, the minute the House issued an arrest warrant, and before any arrest could be made, the target would get an injunction preventing the Sergeant-at-Arms from taking them into custody. Then the case would wind its way through the courts and because, unlike some of the other impeachment-related cases the courts are dealing with now, this would be a "case of first impression," with little precedent to guide it and would probably require extensive and lengthy litigation and the U.S. Supreme Court to resolve.
One of the main issues in such a case would be whether the Sergeant-at Arms/Capitol Police have the jurisdiction to arrest anyone outside of the Capitol complex. Some people here have pointed to the fact that the Supreme Court upheld an inherent contempt arrest in 1937, in a case in which someone was arrested in Ohio by a law officer deputized by the House Sergeant-at-Arms, as proof that the Sergeant-at-Arms can arrest anyone anywhere. But that ruling would have little precedential value in this case, particularly regarding any geographical jurisdiction issue.
First, that case was decided before federal statutes limited Capitol Police geographical jurisdiction to the Capitol complex and immediately surrounding area. In addition, the issue before the Court was "subject-matter jurisdiction," i.e., whether the House had the power to find someone in inherent content. The question of "territorial jurisdiction" was not at issue and not decided by the Court. It's possible - and actually very probable - that the Court today would find that the Speaker of the House can't order the Sergeant-at-Arms to go out and arrest people who aren't physically located on Capitol property.
But regardless how the Court eventually ruled, the case would be tied up in litigation for months, perhaps even years, and no one would be arrested in the meantime.
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