Mainiero's generalized GHZ Poincaré polynomial conjecture

Let $n\geq 3$ and let $\vec{d}$ be a dimension vector. Write $P^{\mathrm{GHZ}}_n(y)=P\left(\lvert\mathrm{GHZ}_{n,\vec{d}}\rangle;y\right)$. Mainiero's generalized GHZ Poincaré polynomial conjecture. For all $n\geq 3$ and dimension vectors $\vec{d}$,

$$P^{\mathrm{GHZ}}_n(y)=1+y^{n-2}+\frac{2}{y}\left((1+y)^n-(1+y^n)\right)$$

and equivalently

$$P^{\mathrm{GHZ}}_n(y)=(2n+1)+2\sum_{m=2}^{n-2}\binom{n}{m}y^{m-1}+(2n+1)y^{n-2}.$$

The coefficients are symmetric, as required by Hodge duality, with linear growth in $n$ at the ends and binomial growth in the interior. The formula was motivated by numerical evidence and the modified Pascal-triangle structure described by Mainiero et al.; its resolution is not specified here.