The proportionality conjecture for the sums $P_{n,t}$ and $A_n$

Let $g$ be the genus, let $n\geq 2$, and let $t\in\{0,\dots,n\}$. For integers $a_1,\dots,a_n\geq 0$ satisfying

$$a_1+\cdots+a_n=2g-3+n,$$

define $P_{n,t}$ by the preceding combinatorial sum and set

$$A_n=(-1)^n(2g-4+n)!\sum_{\substack{o_1,\dots,o_n\in(2\mathbb{Z}+1)_{>0}\o_1+\cdots+o_n=2g-4+n}}\prod_{j=1}^n\binom{2a_j}{o_j}.$$

Proportionality conjecture. For every such $n$, $t$, and $(a_1,\dots,a_n)$, one has

$$P_{n,t}=(-1)^t\left[\left(\binom{n-1}{t}-\binom{n-1}{t-1}\right)(2g-3+n+t)+2(t-1)\binom{n-1}{t-1}\right]A_n.$$

This identity is a proposed combinatorial proportionality relating the quantities obtained by collecting terms according to the number of parity decreases to the basic odd-indexed sum $A_n$. The supplied source does not state whether the conjecture has been proved or refuted, so its resolution remains open.