Cohomological dimension conjecture for the Hodge–Tate stack of a regular ring

Let $R$ be a $p$-complete noetherian regular local ring with perfect residue field. Let $G$ be the group scheme identified in the source with the Hodge–Tate stack presentation $\operatorname{WCart}_{\operatorname{Spf}(R)}^{\mathrm{HT}}\simeq BG$. Cohomological dimension conjecture. The functor

$$\operatorname{R\Gamma}(\operatorname{WCart}_{\operatorname{Spf}(R)}^{\mathrm{HT}},-)$$

or equivalently $\operatorname{R\Gamma}(BG,-)$ carries $\mathcal{D}^{\leq 0}$ to $\mathcal{D}^{\leq\dim(R)}$. The conjecture expresses the expectation that a regular ring is formally smooth from the Hodge–Tate, or “over $\mathbf{F}_1$”, perspective. It is known in dimension one by the preceding results, while the general-dimensional assertion remains open.