Strong eigencone functoriality along folds

Let $G$ be a simple, simply connected algebraic group over $\mathbb{C}$, and let $M\subseteq G$ be a semisimple subgroup that is the fixed subgroup of an automorphism of $G$. Let $M_1$ be a simple factor of $M$ whose highest root coincides with the highest root of $G$. Write $\phi:\overline{\Gamma}_{n}(M_1)\to\overline{\Gamma}_{n}(G)$ for the induced map, and call $\vec{\lambda}\in\overline{\Gamma}_{n}(M_1)$ a point along a fold when

$$\phi^{-1}(\phi(\vec{\lambda}))=\{\vec{\lambda}\}.$$

Strong eigencone functoriality along folds. Along the folds of $\phi$, one has

$$\vec{\lambda}\in\overline{\Gamma}_{n}(M_1)\quad\Longleftrightarrow\quad\vec{\lambda}\in\overline{\Gamma}_{n}(G).$$

This conjecture proposes that the eigencones agree along the parts of the map where no points are identified. The source reports that the conjecture remains open for fixed subgroups of inner automorphisms in types $\operatorname{E}_6$, $\operatorname{E}_7$, and $\operatorname{E}_8$.