The trichotomy conjecture for the cohomology of simple modules under the linearisation map

Let $A\cong \Bbbk Q/I$ be an NCCR of the ring $R$ satisfying Assumption~, and suppose $v_0=1$ for some $0\in Q_0$. Set $X=\mathcal{M}_\theta(A,v)$ for $\theta\in C_+$. For each vertex $i\neq 0$, consider the image $\Psi(S_i)$ under the linearisation map and its cohomology sheaves.

Trichotomy conjecture. For every vertex $i\neq 0$, there is a unique $k(i)\in\\{-1,0\\}$ such that

$$\mathcal{H}^{k(i)}(\Psi(S_i))\neq 0.$$

Moreover, $k(i)=0$ if and only if the locus

$$\\{\theta\in\Theta\mid \theta_i=0\\}$$

is a supporting hyperplane of $C_+$.

This conjecture predicts that the nonzero cohomology of each $\Psi(S_i)$ occurs in exactly one of two adjacent degrees, with the degree determined by whether the corresponding coordinate locus supports the chamber $C_+$. The supplied text does not state a resolution or provide evidence establishing the conjecture, so its status remains open.