Formal Poincare polynomial conjecture for torus actions

Let $V$ be a projective algebraic variety over $k$ with an action of a torus $T$ such that the fixed-point set $V^{T}$ and the set $V^{T,1}$ of one-dimensional $T$-orbits are finite. Let $\Gamma$ be the graph whose vertices are $V^{T}$ and whose edges correspond to one-dimensional $T$-orbits. For a total order $\mathfrak{o}$ on the vertices, orient each edge from the greater vertex to the smaller one, and let $n^{\mathfrak{o}}_{v}$ be the number of arrows with source $v$. Define

$$b^{\mathfrak{o}}_{2i}=\#\{v\in\Gamma:n^{\mathfrak{o}}_{v}=i\},\qquad P^{\mathfrak{o}}(t)=\sum_i b^{\mathfrak{o}}_{2i}t^{2i}.$$

Let $P_V(t)$ be the Poincare polynomial of $V$.

Formal Poincare polynomial conjecture.

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

where $\mathfrak{o}$ ranges over all total orders on the vertices of $\Gamma$.

This conjecture seeks to recover the ordinary cohomology of a cohomologically pure projective variety from the fixed points and one-dimensional orbits of a torus action. Its general status is open.