$P=PP$ or $P=PSPACE$ vs $VP=VNP$
Note that $\#P\in FP^{PP}$ follows from a simple binary search. Given some non-deterministic machine $M$, the language $\{(x,k) |\#M(x)\ge k\}$, where $\#M(x)$ is the number of accepting paths on input $x$, is in PP (you can change the constant in the definition of PP to any FP function). Thus, using a PP oracle we can count the number of accepting paths via binary search.
This implies that $\mathsf{P=PP}$ puts permanent in $VP$, as the problem of counting the number of perfect matchings in a bipartite graph (or equivalently, computing the permanent of the adjacency matrix) is in $\#P$.