The irreducibility conjecture for moduli of nodal curves on K3 surfaces

Let $X$ be a K3 surface, let $L$ be a primitive polarization of genus $g$, and let $\mathcal{V}^n_{g,k}$ denote the moduli space of stable maps from smooth curves to $X$ that are unramified, birational onto their image, and whose image is a nodal curve in $|kL|$ with exactly $n$ nodes. Irreducibility conjecture. The moduli space $\mathcal{V}^n_{g,k}$ is irreducible. This conjecture concerns the geometry of families of nodal curves on polarized K3 surfaces and is described in the source as a deep conjecture that is currently difficult; special cases have been studied.