Equivalence of the matrix factorization categories for the six-line hourglass

Let $\mathrm{X}_{1.9}$ and $\mathrm{X}'_{1.9}$ be the hourglass varieties, and for each $i$ let $\mathcal{MF}_{i}(\mathrm{X}_{1.9})$ denote the left orthogonal component of the exceptional collection $\langle\mathcal{O}_{L_i}(-1),\mathcal{O}_{\widetilde{\mathrm{X}}_{1.9}},\mathrm{U}_{\widetilde{\mathrm{X}}_{1.9}}^{\vee}\rangle$ in $D^b(\mathrm{Coh}(\widetilde{\mathrm{X}}_{1.9}))$. Let $\mathcal{MF}(\mathrm{X}'_{1.9})$ be the matrix factorization category associated with $\mathrm{X}'_{1.9}$. The authors' conjecture. For any $i$, $\mathcal{MF}_{i}(\mathrm{X}_{1.9})$ is equivalent to $\mathcal{MF}(\mathrm{X}'_{1.9})$. This predicts that the residual category is independent of the chosen line. The proposition immediately preceding the conjecture establishes only an isomorphism of $K_0$ groups, so the categorical equivalence remains unproved in the supplied text.