Smoothness conjecture for the determinantal curve associated with nodal elliptic curves

Let $X_0$ be a general hyperelliptic K3 surface, and let $Z_{2r}$ be the determinantal curve constructed to dominate the curve $\Gamma_{2r}$ parametrizing divisors in $|\mathcal{O}(1,r)|$ tangent to the branch curve at $2r$ points. Smoothness conjecture. $Z_{2r}$ is smooth for a general hyperelliptic K3 surface. In particular, the compactified Severi curve satisfies

$$\operatorname{genus}\left(\overline{V}^{L,g-1}\right) \geq O(e^{Cg}).$$

This conjectural smoothness statement would give an exponential lower bound for the genus of the compactified Severi curves, strengthening the preceding lower-bound result for their arithmetic genus.