Existence of a twisted scheme for residual categories of quadric surface bundles

Let $p\colon \mathcal{Q}\to S$ be a quadric surface bundle. Write $S_2$ and $S_3$ for the loci where the fibers have corank at least $2$ and $3$, respectively, and let $\mathcal{A}_{\mathcal{Q}}$ denote its residual category. Residual-category conjecture. If $p$ has simple degeneration generically and every fiber has corank at most $2$, equivalently $S_2\neq S$ and $S_3=\emptyset$, then there is a twisted scheme $(Y,\alpha_Y)$ such that

$$\mathcal{A}_{\mathcal{Q}}\cong \mathbf{D}^{\mathrm{b}}(Y,\alpha_Y).$$

For simple degeneration, the residual category is known to be equivalent to a twisted derived category of the discriminant double cover; the conjecture predicts an analogous twisted-scheme description when corank-two fibers occur, despite the more complicated singularities and reducible Hilbert schemes of lines.