Uehara's autoequivalence-group conjecture for surfaces

Let $S$ be a smooth projective complex surface with $K_S\not\equiv 0$ that admits no minimal elliptic fibration. Let $Z$ be the union of all $(-2)$-curves, and let $B_Z\subseteq\operatorname{Aut}(\operatorname{D}^b(S))$ be the subgroup generated by spherical twists along objects supported on $Z$. Uehara's conjecture. The group of autoequivalences satisfies

$$\operatorname{Aut}(\operatorname{D}^b(S))=\langle B_Z,\operatorname{Pic}(S)\rangle\rtimes\operatorname{Aut}(S)\times\mathbb{Z}[1].$$

This conjecture describes the autoequivalence group in Uehara's case $N_S=2$, separating spherical twists, tensoring by line bundles, automorphisms, and shifts. The supplied text gives no resolution status.