Existence of sheaves realizing connected seminormal partial normalizations

Let $\overline J_{\mathrm{C}}$ be a fine compactified Jacobian of a reduced and projective connected curve $\mathrm{C}$ with planar singularities. Let $\mathrm{C}^{\underline{\nu}}$ be a connected seminormal partial normalization of $\mathrm{C}$ that is maximal with these properties; more generally, it may be any connected partial normalization of $\mathrm{C}$. The curve $\mathrm{C}^{\mathcal I}$ associated with a sheaf $\mathcal I$ is the partial normalization determined by that sheaf.

Existence conjecture. There exists a sheaf $\mathcal I\in\overline J_{\mathrm{C}}$ such that

$$\mathrm{C}^{\mathcal I}=\mathrm{C}^{\underline{\nu}}.$$

If true, this would make the regularity criterion for relative fine compactified Jacobians sharp: the transversality condition would not only suffice but also be necessary. The statement is presented as a conjecture in the source, and no resolution is supplied there.