Semiorthogonal decomposition conjecture for Fano schemes of lines on intersections of two quadrics

Let $X=Q_1\cap Q_2\subset\mathbb{P}^{n+2}$ be a smooth complete intersection of two quadrics in an odd-dimensional projective space. Let $C$ be the hyperelliptic curve associated with the pencil of quadrics, branched over its singular quadrics, and let $F_1(X)$ denote the Fano variety of lines on $X$. Semiorthogonal decomposition conjecture. There is a semiorthogonal decomposition

$$\mathrm{D}^{\mathrm{b}}(F_1(X))=\left\langle\mathrm{D}^{\mathrm{b}}(\operatorname{Sym}^2 C),\underbrace{\mathrm{D}^{\mathrm{b}}(C),\dots,\mathrm{D}^{\mathrm{b}}(C)}_{n-3\text{ times}},\underbrace{\mathrm{D}^{\mathrm{b}}(\mathbb{C}),\dots,\mathrm{D}^{\mathrm{b}}(\mathbb{C})}_{{n-4\choose 2}+2(n-4)\text{ times}}\right\rangle.$$

This is the specialization of a broader conjecture on semiorthogonal decompositions for Fano schemes of linear spaces on intersections of quadrics. The paper uses it as input for the subsequent orthogonal-Grassmannian conjecture; its resolution status is not specified in the supplied text.