Craw's tilting bundle and moduli conjecture for smooth toric Fano varieties

Let $X$ be a smooth toric Fano variety. A sequence of basepoint-free line bundles is given by

$$\mathcal{L}=(\mathscr{O}_X,L_1,\dots,L_r).$$

Assume that this sequence has bound quiver of sections $(Q,\varrho)$, and let $\operatorname{Wt}(Q)$ denote the weights of $Q$. Craw's conjecture. There is such a sequence for which

$$X\cong\mathcal{M}_\vartheta(Q,\varrho)$$

for the weight $\vartheta\in\operatorname{Wt}(Q)$ satisfying $\vartheta_i>0$ for $i\neq 0$, and for which the tautological bundle

$$\bigoplus_{0\leq i\leq r}L_i$$

is tilting, giving the derived equivalences stated in the source. This conjecture is motivated by constructions of tilting bundles on smooth toric Fano threefolds and fourfolds; the source does not provide a resolution status, so its general validity remains open.