Regularity criterion via the Hodge–Tate stack

Let $X$ be a noetherian excellent $p$-adic formal scheme, and write $\operatorname{WCart}_X^{\mathrm{HT}}$ for its derived Hodge–Tate stack, namely the Hodge–Tate locus in $\operatorname{WCart}_X$. Let $\pi:\operatorname{WCart}_X^{\mathrm{HT}}\to X$ be the Hodge–Tate structure map. Regularity criterion conjecture. The following are equivalent: $X$ is regular; $\pi$ is a gerbe for a $p$-completely flat $X$-group scheme, so that $\operatorname{WCart}_X^{\mathrm{HT}}$ and $\operatorname{WCart}_X$ are classical; and there exists an integer $N$ such that

$$\mathrm{Ext}^{>N}_{\operatorname{WCart}_X^{\mathrm{HT}}}(-,-)$$

vanishes on $p$-torsion quasi-coherent sheaves on $\operatorname{WCart}_X^{\mathrm{HT}}$. This is presented as a strong converse to the cohomological-dimension conjecture for regular rings. Its equivalence of regularity, classical gerbe structure, and bounded Ext-dimension remains open.