you can't wave your hands, say "zero knowledge", and be private. this is too little information to be useful

usually, the implication of ZKP is that you buy coupons and claim them without attribution. in this coupon scenario the ZKP can just be a blind signature scheme.

however signal has an obscene fondness for TEEs (secure enclaves) so they may actually be doing something stupid here which will require trust beyond the ZKP.