Asymptotic irreducibility conjecture for universal Severi varieties

Fix an integer $g\geqslant 2$, let $\mathcal{M}^{\circ}_{K3,g}$ be the indicated moduli space of polarized K3 surfaces, and let $\mathcal{V}_{k,h}$ be the universal Severi variety associated with $|kL_m|$. Its fibers have arithmetic genus

$$p_a(k)=1+k^2(g-1).$$

Asymptotic irreducibility conjecture. For every $\varepsilon>0$, if $k$ is sufficiently large with respect to $\varepsilon$, then for every integer $h$ satisfying

$$\varepsilon p_a(k)\leqslant h\leqslant p_a(k),$$

the universal Severi variety $\mathcal{V}_{k,h}$ is irreducible.

This is a weaker version of the conjecture that all universal Severi varieties are irreducible, but it is stated as sufficient for the paper's purposes. It guarantees irreducibility only when the geometric genus is at least an $\varepsilon$-fraction of the arithmetic genus, and the source gives no resolution.