Splitting conjecture for the Langlands-Shahidi derived category

Let $G$ be a quasi-split reductive group over $E$, let $\operatorname{Bun}_G$ be its moduli stack of $G$-bundles, and let $\operatorname{Bun}_G^b$ denote the stratum indexed by $b\in\lvert\operatorname{Bun}_G\rvert$. Let $i_b\colon\operatorname{Bun}_G^b\to\operatorname{Bun}_G$ be the inclusion, and let $\operatorname{\mathcal{D}}^{\mathrm{LSt}}$ denote the Langlands-Shahidi type subcategory. Splitting conjecture. The semi-orthogonal decomposition on $\operatorname{\mathcal{D}}^{\mathrm{LSt}}(\operatorname{Bun}_G)$ splits, giving an equivalence

$$\operatorname{\mathcal{D}}^{\mathrm{LSt}}(\operatorname{Bun}_G)\simeq\prod_{b\in\lvert\operatorname{Bun}_G\rvert}\operatorname{\mathcal{D}}^{\mathrm{LSt}}(\operatorname{Bun}_G^b),$$

with the functor induced by the collection of pushforwards $i_{b!}$, equivalently $i_{b*}$ or $i_{b\sharp}$; all three pushforwards are isomorphic in this case. This concerns the decomposition of the Langlands-Shahidi type category by bundle strata. The source provides no evidence of resolution, so the conjecture is open.