Characteristic-zero purity conjecture for K3 unions of scrolls

Let $e=(e_1,e_2)\in \mathbb{F}^2$ with $e_2\neq 0$, and let $X_e(a,a)$ be the corresponding union of scrolls over a field $\mathbb{F}$ of characteristic $0$. Write

$$p(z)=z^2-e_1z+e_2=(z-t_1)(z-t_2).$$

Characteristic-zero purity conjecture. The union of scrolls $X_e(a,a)$ has a pure resolution unless $t_2/t_1\neq 1$ is a $k$-th root of unity for some $k\leq (a+1)/2$.

The conjecture is motivated by determinant computations over $\mathbb{Z}[e_1,e_2]$, whose exceptional factors indicate resonance phenomena. The source presents this as experimental and gives no proof.