Not that I see anywhere. Suppressors are treated differently than SBRs, SBSs, or full auto firearms.
Do I give up any 4th Amd protections?
The ATF has the legal authority to demand to see your tax stamp (also serves as registration document). Suppressors are considered NFA "firearms" as a legal definition.
They don't have the right to see your suppressor, at least as far as I can tell.
It seems highly unlikely they would ask to see the tax stamp if you do not have the suppressor in your possession.
😉👌