The homological-dimension conjecture for Kuznetsov components of prime Fano threefolds

Let $X$ be a cubic threefold, quartic double solid or Gushel--Mukai threefold, and let $\mathcal{K}u(X)$ be its Kuznetsov component. Let $\sigma$ be a stability condition on $\mathcal{K}u(X)$, and write $\mathrm{homdim}(\sigma)$ for its homological dimension. Homological-dimension conjecture. For every stability condition $\sigma$ on $\mathcal{K}u(X)$,

$$\mathrm{homdim}(\sigma)\leq 2.$$

The paper proves that homological dimension at most $2$, global dimension at most $2$, and Serre invariance are equivalent for these three types of Kuznetsov components. The conjecture would therefore describe all stability conditions and, together with the contractibility result for the Serre-invariant component, imply $\operatorname{Stab}(\mathcal{K}u(X))\cong\widetilde{\mathrm{GL}}^+(2,\mathbb{R})\cong\mathbb{C}\times\mathbb{H}$; it remains open in the source.