Irreducibility conjecture for universal Severi varieties of curves on K3 surfaces

Let $\mathcal{B}_g$ be the moduli stack of primitively polarized K3 surfaces of genus $g>2$, parametrizing pairs $(S,M)$ where $M$ is ample and primitive with $(M\cdot M)=2g-2$. Let $\mathcal{U}\subseteq\mathcal{B}_g$ be the open substack where the Severi variety $V_h(S,M)$ is nonempty, and let $\mathcal{V}^g_{h,k}$ be the stack over $\mathcal{U}$ whose fibre over $(S,M)$ is $V_h(S,kM)$. Irreducibility conjecture. The universal Severi variety $\mathcal{V}^g_{h,k}$ is irreducible. This concerns the global irreducibility of Severi varieties in varying primitively polarized K3 surfaces; smoothness is known, while irreducibility is posed as a conjecture in the source.