Conjecture on graded local Weyl modules and support-independent dimension

Let $\lambda\in P_0^+$, where $P_0^+$ denotes the set of dominant weights relevant to the twisted current algebra, and let a graded local Weyl module and the associated graded module of the restriction of a local Weyl module for $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$ be given. Graded local Weyl module conjecture. The graded local Weyl module is isomorphic to the associated graded module of the restriction of a local Weyl module for $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$. Moreover, the dimension of a local Weyl module of highest weight $\lambda$ is independent of the support of the module. The conjecture addresses the unresolved structure and dimension of local Weyl modules in the even case $\lambda(\alpha_l^{\vee})$; in this setting, the identification with Demazure modules fails and the dimension and character remain uncomputed.