The rational-tail and nonseparating-boundary conjecture for geometric interpretations of type-A conformal blocks

Let $\mathbb{V}$ be a vector bundle of conformal blocks of type $A$ on $\overline{\mathcal{M}}_{g,n}$. For each $x\in Z$, let $(\mathcal{X}_x,\mathcal{L}_x)$ be a polarized pair, and let $\mathcal{M}^{\mathrm{rt}}_{g,n}$ denote the rational-tail locus and $\Delta^0_{\mathrm{irr}}$ the specified open part of the irreducible boundary divisor. Set

$$Z=\mathcal{M}^{\mathrm{rt}}_{g,n}\cup\Delta^0_{\mathrm{irr}}.$$

Geometric interpretation conjecture. There are polarized pairs $(\mathcal{X}_x,\mathcal{L}_x)$ such that the isomorphism in Equation holds for every $x\in Z$.

This conjecture proposes a precise locus where conformal-block bundles of type $A$ admit geometric interpretations. It is motivated by examples for genera at most $3$; the statement concerns existence on the indicated locus, while the paper's preceding theorem shows that such interpretations cannot hold uniformly on all of $\overline{\mathcal{M}}_{g,n}$ in general.