Moment-graph formula conjecture for the Poincaré polynomial of a fundamental domain

Let $F_{\gamma}$ be the fundamental domain equipped with the action of $\widetilde{T}=T\times \mathbb{G}_{m}$, and let $\Gamma$ be its moment graph, with vertices $F_{\gamma}^{\widetilde{T}}$ and edges $F_{\gamma}^{\widetilde{T},1}$. For a total order $\mathfrak{o}$ on the vertices of $\Gamma$, orient every edge so that its source is greater than its target. If $n^{\mathfrak{o}}_{v}$ is the number of arrows with source $v$, define

$$b_{2i}^{\mathfrak{o}}=\sharp\{v\in\Gamma:n^{\mathfrak{o}}_{v}=i\},$$

and

$$P^{\mathfrak{o}}(t)=\sum_i b_{2i}^{\mathfrak{o}}t^{2i}.$$

Let $P(t)$ be the Poincaré polynomial of $F_{\gamma}$.

Moment-graph formula conjecture. One has

$$P(t)=\min_{\mathfrak{o}}\{P^{\mathfrak{o}}(t)\},$$

where $\mathfrak{o}$ ranges over all total orders on the vertices of $\Gamma$, and the minimum is taken with respect to the ordering in which $P_1(t)<P_2(t)$ when the leading coefficient of $P_2(t)-P_1(t)$ is positive.

This conjectural formula would recover the Poincaré polynomial of the fundamental domain from its moment graph. The statement depends on the cohomological purity framework described immediately before it, but the source gives no resolution or further evidence for the conjecture.