Finiteness conjecture for curve components of derived categories of threefolds

Let $X$ be a threefold satisfying the stated properties, including a decomposition of its intermediate Jacobian

$$J(X)=\bigoplus_{i=1}^N J(S_i),$$

where the $S_i$ are smooth projective curves of positive genus and the associated fully faithful functors $D^b(S_i)\to D^b(X)$ exist. Let $C_{D^b(S_i)}^{{\rm Aut}(D^b(X))}(D^b(X))$ denote the set of such curve components modulo the action of the group of exact autoequivalences of $D^b(X)$. Finiteness conjecture. For each $i=1,\dots,N$, one has

$$\left\lvert C_{D^b(S_i)}^{{\rm Aut}(D^b(X))}(D^b(X))\right\rvert<\infty.$$

The claim is proposed for the listed examples of threefolds, including smooth quadrics, certain projective bundles and Fano threefolds, complete intersections of two quadrics, rational conic bundles, and the specified Del Pezzo fibrations. The conjecture concerns finiteness after quotienting by all derived autoequivalences, whereas the preceding result identifies the possible positive-genus curves with the factors $S_i$ of the intermediate Jacobian.