A query regarding P vs NP
Your problem is essentially a $NP^{coNP}$ problem. If $P=NP$ then the polynomial hierarchy collapses and your problem is in $P$.

If $P=NP=coNP$ then we can check in polynomial time whether a solution $S_0$ is valid (using a polynomial-time algorithm to check each property, which exists because $coNP=P$). Thus, the problem is in $NP$, and thus, in $P$.