Hilbert-scheme degeneration conjecture for quartic double solids

Let $Y\to\mathbb{P}^3$ be a quartic double solid whose ramification divisor contains no lines, and let $\cX/B$ be the family of prime Fano threefolds of genus $6$ constructed in Theorem 1.1. Hilbert-scheme degeneration conjecture. There exists a stability condition $\underline{\upsigma}$ on $\bcA_{\cX}$ and a section $\bv\in\operatorname{K}_0^{\mathrm{num}}(\bcA_{\cX}/B)(B)$ such that the associated moduli space $\rM_{\underline{\upsigma}}(\bcA_{\cX},\bv)$ is a smooth and proper family of surfaces $\cF(\cX/B)$ with fibers $\cF(\cX/B)_b\cong\rF^{\mathrm{min}}_2(\cX_b)$ for $b\neq o$ and $\cF(\cX/B)_o\cong\rF_1(Y)$. This predicts a categorical interpolation between the minimal model of the Hilbert scheme of conics on the general Fano threefold and the Hilbert scheme of lines on the quartic double solid.