Generalized affine-paving conjecture for cohomologically pure varieties

Let $V$ be a projective algebraic variety over $k$ admitting an action of a torus $T$ such that the fixed points $V^{T}$ and the one-dimensional orbits $V^{T,1}$ are finite. For a total order $\mathfrak{o}$ on the vertices of the associated graph $\Gamma$, let $V^{\mathfrak{o}}(v)$ be the locally closed pieces obtained inductively from the attracting sets of the vertices. A generalized affine paving is a decomposition into locally closed subvarieties having the same compactly supported cohomology as an affine space $\mathbf{A}^{n}$.

Generalized affine-paving conjecture. If $\mathfrak{o}$ is a total order such that $P^{\mathfrak{o}}(t)=P_V(t)$, then the decomposition of $V$ according to $\mathfrak{o}$ is a generalized affine paving.

The conjecture proposes a general method for constructing affine-paving-like decompositions from torus fixed-point and one-dimensional-orbit data. Its general status is open.