The strong Schottky problem for Hirota varieties

Let $\mathcal{H}_\mathcal{C}^{M}$ be the main component of the Hirota variety $\mathcal{H}_\mathcal{C}$, and let $\mathcal{H}_\mathcal{C}^{I}$ denote its distinguished irreducible component. Strong Schottky problem. The main component satisfies

$$\mathcal{H}_\mathcal{C}^{M}=\mathcal{H}_\mathcal{C}^{I}.$$

This conjecture strengthens the weak Schottky problem by identifying the main component with the distinguished irreducible component; the supplied text gives no resolution, so the problem remains open.