The D-affinity conjecture for Frobenius pushforwards

Let $X$ be a globally $F$-split variety. Write $D_X^{(e)}$ for the sheaf of $O_X^{p^e}$-linear endomorphisms of $\mathcal O_X$, so that $F_*^eD_X^{(e)}=\operatorname{\mathcal{E}nd}F_*^e\mathcal O_X$. D-affinity conjecture. For $e'>e$, the vector space $H^i(D_X^{(e)})$ is a direct summand of $H^i(D_X^{(e')})$; consequently, if $F_*^e\mathcal O_X$ is not tilting for some $e$, then $X$ is not $D$-affine. This would connect the tilting property of Frobenius pushforwards with $D$-affinity, since a $D$-affine variety must have vanishing higher cohomology for its sheaf of differential operators. The paper presents both assertions as conjectural and does not establish them.