Positive-characteristic purity conjecture for general K3 unions of scrolls

Let $a$ be a positive integer, let $\overline{\mathbb{F}}$ be an algebraically closed field of characteristic $p$, and let $e=(e_1,e_2)\in\overline{\mathbb{F}}^2$ be general. Let $X_e(a,a)$ be the corresponding union of scrolls. Positive-characteristic purity conjecture. The union of scrolls $X_e(a,a)$ has a pure resolution if

$$p\geq a.$$

In particular, Green's conjecture holds for a general curve of genus $g$ over a field of characteristic $p$ if $p\geq (g-1)/2$. This is an experimental characteristic bound inferred from the computed resolutions; the source does not establish it in general.