Autoequivalence conjecture for exceptional objects on the Hirzebruch surface $\mathbb{F}_2$

Let $\mathbb{F}_2$ be the Hirzebruch surface, and let $D(\mathbb{F}_2)$ denote its derived category. An object $\mathcal{E}\in D(\mathbb{F}_2)$ is exceptional if its derived endomorphism algebra is the ground field and it has no higher self-extensions. An exceptional vector bundle is an exceptional object represented by a vector bundle on $\mathbb{F}_2$. Write $\operatorname{Auteq}(D(\mathbb{F}_2))$ for the group of autoequivalences of $D(\mathbb{F}_2)$.

Autoequivalence conjecture. For any exceptional object $\mathcal{E}\in D(\mathbb{F}_2)$, there exists an autoequivalence $\Phi\in\operatorname{Auteq}(D(\mathbb{F}_2))$ such that $\Phi(\mathcal{E})$ is an exceptional vector bundle on $\mathbb{F}_2$.

The conjecture proposes that spherical twists and the other known autoequivalences of $D(\mathbb{F}_2)$ account for all exceptional objects up to taking an exceptional vector bundle to an autoequivalent object. Its status is not determined by the supplied text.