Cao–Jiang conjecture on exceptional objects of weak del Pezzo surfaces

Let $X$ be a weak del Pezzo surface, meaning a smooth projective surface with nef and big anticanonical bundle. Let $D(X)=D^b(\operatorname{Coh} X)$ be its bounded derived category of coherent sheaves, and let an object of $D(X)$ be exceptional if

$$\operatorname{Hom}(\mathcal{E},\mathcal{E}[i])\cong \begin{cases} \mathbb{C} & i=0,\\ 0 & i\ne 0. \end{cases}$$

Write $\operatorname{Auteq}(D(X))$ for the group of autoequivalences of $D(X)$.

Cao–Jiang conjecture. For any exceptional object $\mathcal{E}\in D(X)$, there exists an autoequivalence $\Phi\in\operatorname{Auteq}(D(X))$ such that $\Phi(\mathcal{E})$ is an exceptional vector bundle or a line bundle on a $(-1)$-curve on $X$.

Exceptional objects on del Pezzo surfaces are known to be shifts of exceptional vector bundles or line bundles on $(-1)$-curves. This conjecture proposes that the same classification holds for exceptional objects on weak del Pezzo surfaces after applying autoequivalences, accounting for the spherical twists arising from $(-2)$-curves.