The coherent cohomological dimension conjecture for Shimura varieties

Let $(G,X)$ be a Shimura datum with $G^{ad}$ simple, and let $Sh_K(G,X)$ be the associated Shimura variety of level $K$. Its dimension is denoted by $\dim(Sh_K(G,X))$, and $\operatorname{rank}_{\mathbb{Q}}G^{ad}$ denotes the $\mathbb{Q}$-rank of the adjoint group. The coherent cohomological dimension of a scheme $X$ is

$$\operatorname{ccd}(X)=\max\left\{i\geq 0\mid \exists\mathcal{F}\in\operatorname{Coh}(X),\ H^i(X,\mathcal{F})\neq 0\right\}.$$

The coherent cohomological dimension conjecture. One conjectures that

$$\operatorname{ccd}(Sh_K(G,X))\leq \dim(Sh_K(G,X))-\operatorname{rank}_{\mathbb{Q}}G^{ad}.$$

This is presented as a general conjecture extending the preceding prediction for Siegel modular varieties. The paper does not establish the inequality in this generality.