Functoriality conjecture for Kloosterman sheaves

Let $G$ and $G'$ be split almost simple groups over $k$ whose dual groups satisfy $\check{G}\supset\check{G}'$ and appear in the same line of the paper's global-monodromy table. Let $\phi$ be a generic linear function for $G$. Functoriality conjecture. There exists a generic linear function $\phi'$ for $G'$ such that, over $U={\mathbb P}^1_{\backslash\{0,\infty\}}\otimes_k\overline{k}$, the $\check{G}$-Kloosterman sheaf $\operatorname{Kl}_{\check{G}}(\phi)$ is the pushout of the $\check{G}'$-Kloosterman sheaf $\operatorname{Kl}_{\check{G}'}(\phi')$. This predicts compatibility of these sheaves under the indicated dual-group inclusion and is proposed as a further direction after the paper's monodromy calculations.