The deformation conjecture for Frobenius pushforwards on smooth projective varieties

Let $X$ be a smooth projective variety with nontrivial first-order deformations, meaning $h^1(T_X)>0$, where $T_X$ is the tangent sheaf. The absolute Frobenius pushforward is denoted by $F_*^e$. Deformation conjecture. Then $F_*^e\mathcal O_X$ is not tilting for any $e>0$. This conjecture predicts that nontrivial infinitesimal deformations obstruct the tilting property of every Frobenius pushforward of the structure sheaf. The paper discusses this as a generalization of the preceding examples, but provides no proof or resolution.