Quantum folding conjecture for admissible diagram automorphisms

Let $\mathfrak{g}$ be a Lie algebra, let $\sigma$ be an admissible diagram automorphism, and let $\mathfrak{g}_+$, $\mathfrak{n}$, $R(w_\circ)$, $\iota_\mathbf{i}$, and $\tilde\iota_\mathbf{i}$ denote the folded objects associated with this data. Assume that ${\mathfrak{g}^\sigma}^\vee$ has no Lie ideals of type $G_2$. A quantum folding conjecture asserts that there exists a unique $\mathfrak{g}_+$-module $V_\mathfrak{g}$ such that, for every $\mathbf{i}\in R(w_\circ)$, the folding $\iota_\mathbf{i}$ is tame enhanced liftable, the enhanced uberalgebra $\hat U(\iota_\mathbf{i})$ is a flat deformation of both $U(\mathfrak{n}\rtimes\mathfrak{g}_+)$ and $S(V_\mathfrak{g}\rtimes\mathfrak{g}_+)$, and

$$\mathscr{Frac}(U(\iota_\mathbf{i}))$$

is generated by $\tilde\iota_\mathbf{i}(E_r)$ for $r\in I/\sigma$, where $E_r$ are Chevalley generators of $U_q^+({\mathfrak{g}^\sigma}^\vee)$ and $\tilde\iota_\mathbf{i}:U_q^+({\mathfrak{g}^\sigma}^\vee)\hookrightarrow U(\iota_\mathbf{i})$ is the lifting of $\iota_\mathbf{i}$. This conjecture proposes a uniform structure for the tame quantum foldings covered by the preceding theorems; the exclusion of Lie ideals of type $G_2$ reflects the computational and structural difficulties encountered in the exceptional case. The supplied text gives no evidence that the conjecture has been resolved.