The zonotope criterion for generating line bundles on smooth toric varieties

Let $X$ be a smooth toric variety, and let $Z$ be the zonotope appearing in the source. Suppose the positive-dimensional subvarieties admitting linear inclusions into $X$ are

$$Y_0=X,Y_1,\ldots,Y_k.$$

For divisors $D_0,\ldots,D_k$ on $X$, assume that $\mathcal O_X(D_j)\in Z$ and that the corresponding linear inclusions satisfy

$$\operatorname{Hom}_{D^b\operatorname{Coh}(X)}^\bullet(\mathcal O_{Y_j},\mathcal O_X(D_j))\neq 0$$

for every $j$.

Zonotope generation conjecture. Under these assumptions,

$$\bigoplus_{j=0}^k\;\bigoplus_{\mathcal F\in\operatorname{Frob}(D_j)}\mathcal F$$

is a classical generator of $D^b\operatorname{Coh}(X)$.

The conjecture is intended to imply the preceding Frobenius-generation conjecture: the preceding geometric results reduce possible nonzero-cohomology obstructions to line bundles over vertices of the relevant zonotope, while this statement predicts that the finite collection of Frobenius summands selected by the $D_j$ generates.