Isomorphism conjecture for local and graded local Weyl modules

Let $\mathfrak a$ and $\mathfrak b$ be such that $U_{\mathbb Z}(\mathfrak a)$ and $U_{\mathbb Z}(\mathfrak b)$ have been defined. Let $\mathbf a=(a_1,\dots,a_n)\in\mathbb F^n$, and let $\varphi_{\mathbf a}$ be the Lie algebra automorphism of $\mathfrak g[n]_{\mathbb F}$ given by

$$x\otimes f(t_1,\dots,t_n)\mapsto x\otimes f(t_1-a_1,\dots,t_n-a_n).$$

Write $\operatorname{res}(W_{\mathbb F}(\pmb\omega^{\lambda,\mathbf a}))$ for the module obtained by restricting the action of $U_{\mathbb F}(\mathfrak g\langle n\rangle)$ to $U_{\mathbb F}(\mathfrak g[n])$, and let $\varphi_{\mathbf a}^*(W_{\mathbb F}(\pmb\omega^{\lambda,\mathbf a}))$ be its pull-back by $\varphi_{\mathbf a}$. Isomorphism conjecture for local and graded local Weyl modules. For any $\mathbf a\in\mathbb F^n\setminus\{\mathbf 0\}$, $\varphi_{\mathbf a}^*(W_{\mathbb F}(\pmb\omega^{\lambda,\mathbf a}))$ is isomorphic to $W_{\mathbb F}^n(\lambda)$.

This conjecture proposes a generalization to hyper current and hyper loop algebras of the known relationship between local Weyl modules and graded local Weyl modules in the case $n=1$. The source explains that the standard proof does not extend for $n>1$ because Demazure modules, lattice constructions, and field-independent dimension results are unavailable; the status is therefore left open.