Characteristic-independent Cohen–Macaulay conjecture for strongly critical weights

Let $G$ be a reductive group with maximal torus $H$ acting on a representation $W$ with weights $\beta_1,\ldots,\beta_d\in X(H)$. Let

$$\Sigma=\left\{\sum_{i=1}^d a_i\beta_i\mid a_i\in(-1,0]\right\}\subset X(H)_{\mathbb{R}},$$

and let $\rho$ be half the sum of the positive roots. A weight $\alpha\in X(H)^+\cap(-2\rho+\Sigma)$ is strongly critical, and $T(\alpha)$ is the tilting module replacing the simple module $L(\alpha)$. Characteristic-independent Cohen–Macaulay conjecture for strongly critical weights. Theorem 1.3 holds in arbitrary characteristic provided $\operatorname{Sym}(W)$ has a good filtration and $L(\alpha)$ is replaced by $T(\alpha)$; namely, under these conditions, $M(T(\alpha)^*)$ is Cohen–Macaulay for strongly critical $\alpha$. The conjecture is presented as evidence-based and reasonable, extending a characteristic-zero theorem to arbitrary characteristic under a good-filtration hypothesis. Its resolution status is not given in the supplied text.