Has it been proven that the optimization TSP is (or is not) polynomial-time verifiable if P ≠ NP?
If $optTSP$ is in $NP$, then $coNP = NP$.  The latter is unresolved currently.

Proof: if $optTSP$ is in $NP$, then $coTSP$ is in $coNP$ (the certificate being whatever certificate was provided to verify a solution to $optTSP$ was minimal, and then comparing the value of that solution to the desired bound). And because $TSP$ is $NP$-complete then this implies $NP = coNP$.