Local genus polynomials are log-concave

Let $(G,v)$ be a rooted graph. For an almost complete rotation system at $v$, let the local genus polynomial be the genus polynomial of the embeddings consistent with that system. Local log-concavity conjecture. All the local genus polynomials of every rooted graph $(G,v)$ are log-concave. This is the local form of the conjecture that genus distributions are log-concave; its equivalence with the CLLC formulation is stated in the source, while the general claim is presented as unresolved here.