Vertically symmetric nest distribution conjecture

Let $(a)_k=\Gamma(a+k)/\Gamma(a)$, let $G=G^{\rm V}_{2n}$ be the vertically symmetric fully packed loop grid, and let $P_G(k)$ denote the number of diagrams with $k$ nests. Define

$$A^{\rm V}_{2n+1}=\prod_{j=0}^{n-1}(3j+2)\frac{(6j+3)!(2j+1)!}{(4j+2)!(4j+3)!}=1,3,26,646,\ldots.$$

Vertically symmetric nest distribution conjecture. The nest distribution is

$$P_G(k)=k\frac{4^{n+k}}{27^n}\frac{(1/2)_{n+k}}{(1/3)_{2n}}\frac{(3n+1)!(2n-k-1)!}{n!(n-k)!(2n+k+1)!}A^{\rm V}_{2n+1}.$$

Here $A^{\rm V}_{2n+1}$ is the number of vertically symmetric $(2n+1)\times(2n+1)$ ASMs. The conjecture was checked for $n$ up to $8$, but remains unresolved in the supplied text.