The irreducibility conjecture for universal Severi varieties of K3 surfaces

Let $\mathcal F_g$ be the irreducible $19$-dimensional moduli stack of primitively polarized $K3$ surfaces of genus $g$, and let $\mathcal{V}_{g,\delta}$ be the universal Severi variety parametrizing $\delta$-nodal curves on the corresponding polarized surfaces, for $0\leq \delta\leq g$.

Universal Severi irreducibility conjecture. For every $0\leq \delta\leq g$, the universal Severi variety $\mathcal{V}_{g,\delta}$ is irreducible.

This is a weaker form of the fiberwise Severi irreducibility conjecture and was considered more approachable. The source presents it as a conjectural statement; no resolution is supplied in the given text.