Orthogonal Grassmannian decomposition conjecture for a pencil of quadrics

Let $\langle Q_1,Q_2\rangle$ be a pencil of quadrics in $\mathbb{P}^{n+2}$ whose base locus is smooth in an odd-dimensional projective space. Let $C$ be the hyperelliptic curve associated with the pencil, and let $\operatorname{OGr}(2,\mathfrak{Q})$ denote the relative orthogonal Grassmannian of isotropic 2-planes. Pencil conjecture. There is a semiorthogonal decomposition

$$\mathrm{D}^{\mathrm{b}}(\operatorname{OGr}(2,\mathfrak{Q}))=\left\langle\underbrace{\mathrm{D}^{\mathrm{b}}(C),\dots,\mathrm{D}^{\mathrm{b}}(C)}_{n+1\text{ times}},\underbrace{\mathrm{D}^{\mathrm{b}}(\mathbb{C}),\dots,\mathrm{D}^{\mathrm{b}}(\mathbb{C})}_{(n-1)(n+1)\text{ times}}\right\rangle.$$

This conjecture is obtained formally by subtracting the conjectural decomposition for $F_1(X)$ from the known decomposition for the Hilbert square of the base locus. The paper interprets it as a relativization of full exceptional collections for Grassmannians of isotropic lines; no resolution status is supplied.