The linear-obstruction characterization of Frobenius generators for smooth toric varieties

Let $X$ be a smooth toric variety, let $D$ be a divisor on $X$, and let $F_\ell$ denote the toric Frobenius map for some $\ell\in\mathbb N$. A subvariety $Y$ with a linear inclusion $\phi\colon Y\to X$ is understood via its structure sheaf $\phi_*\mathcal O_Y$.

Linear-obstruction conjecture. $(F_\ell)_*\mathcal O_X(D)$ is a classical generator of $D^b\operatorname{Coh}(X)$ for some $\ell\in\mathbb N$ if and only if

$$\operatorname{Hom}^\bullet_{D^b\operatorname{Coh}(X)}(\phi_*\mathcal O_Y,\mathcal O_X(D))\neq 0$$

for every linear inclusion $\phi\colon Y\to X$.

This conjecture asserts that linear morphisms give all obstructions to generation by the Frobenius pushforward of a line bundle. The necessity follows from the preceding linear-obstruction proposition; the sufficiency is the unresolved direction in the source.