The exact-sequence conjecture for autoequivalences of elliptic surfaces

Let $S$ be a smooth projective elliptic surface with $\kappa(S)\ne 0$. Let $B$ be generated by the twist functors $T_{\mathcal{O}_G(a)}$ for $(-2)$-curves $G$, let $F$ be a fiber, and let $J_S(b)=J_S(1,b)$. The group $\Gamma_0(\lambda_S)$ is the congruence subgroup defined by

$$\Gamma_0(\lambda_S)=\left\{\begin{pmatrix}c&a\\d&b\end{pmatrix}\in\operatorname{SL}(2,\mathbb Z)\mathrel{\bigm|}d\in\lambda_S\mathbb Z\right\}.$$

The exact-sequence conjecture. There should be a short exact sequence

$$1\to \left\langle B,\otimes\mathcal{O}_S(D)\mid D\cdot F=0,\;F\text{ is a fiber}\right\rangle\rtimes\operatorname{Aut}(S)\times\mathbb Z[2]\to\operatorname{Auteq}D(S)\xrightarrow{\Theta}\left\{\begin{pmatrix}c&a\\d&b\end{pmatrix}\in\Gamma_0(\lambda_S)\mathrel{\bigm|}J_S(b)\cong S\right\}\to1.$$

Here $\Theta$ is induced by the action of $\operatorname{Auteq}D(S)$ on the even-degree part $H^0(F,\mathbb Z)\oplus H^2(F,\mathbb Z)\cong\mathbb Z^2$ of the integral cohomology of a smooth fiber $F$. The claim would describe all autoequivalences through their cohomological action, with kernel generated by twists, suitable tensor products, automorphisms, and shifts; its status is not resolved by the supplied context.