The irreducibility conjecture for the ordered-node space of rational curves on polarized K3 surfaces

Let ${\mathcal K}_p$ be the moduli space of polarized K3 surfaces $(X,L)$ of genus $p$, and let ${\mathcal V}_{L,0}$ be the universal Severi variety of rational curves in $|L|$ over ${\mathcal K}_p$. Define the space of rational curves with an ordering of their $p$ singular points by

$${\mathcal W}_{L,0}=\left\{(X,L,C,s_1,s_2,\ldots,s_p):(X,L,C)\in{\mathcal V}_{L,0},\ C_{\operatorname{sing}}=\{s_1,s_2,\ldots,s_p\}\right\}.$$

The ordered-node irreducibility conjecture. Then ${\mathcal W}_{L,0}$ is irreducible. This is identified as the difficult monodromy-related part of the proposed approach to the irreducibility of the universal Severi variety; the source does not prove it.