Mainiero's generalized W Poincaré polynomial conjecture

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

$$P^{\mathrm{W}}_n(y)=3+3y^{n-2}.$$

This is the closed-form expression paired in the source with the generalized GHZ formula, and is motivated by numerical evidence for the independence of the polynomial from the local Hilbert-space dimensions. Its resolution is not specified here.