Conjecture on spherical objects of the derived category of a log Calabi–Yau surface

Let $Y$ be the surface under consideration, and let $i:C\hookrightarrow Y$ denote the inclusion of a $(-2)$ curve. Define the collection of spherical objects

$$\mathcal{A}:=\{i_\ast\mathcal{O}_C(a)\mid C\text{ is a }(-2)\text{ curve},\ a\in\mathbb{Z}\}.$$

Let $T_{\mathcal{A}}$ be the group generated by spherical twists in objects of $\mathcal{A}$. Spherical-object conjecture. Every spherical object in $D(Y)$ is quasi-isomorphic to an element of $T_{\mathcal{A}}\cdot\mathcal{A}$. Any spherical object has support in the union $Z$ of all $(-2)$ curves in $Y$; the claim is known for connected components of $Z$ that are chains or cycles of $(-2)$ curves, subject to the technical qualification noted in the source. A full description of the autoequivalence group would require this classification.