Affine Langlands character conjecture for projected simple characters

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra with weight lattice $P$, and let $\Pi$ be the projection from characters of $\mathfrak{g}$ to the Langlands-dual character lattice. Let $L$ be an irreducible finite-dimensional representation of $\mathfrak{g}$, with character $\chi(L)$, and let $U_t({}^L\widehat{\mathfrak{g}})$ be the quantum affine algebra of the Langlands dual affine Lie algebra.

Affine Langlands character conjecture. The polynomial

$$\Pi(\chi(L))$$

is the character of a representation of $U_t({}^L\widehat{\mathfrak{g}})$.

The statement strengthens the corresponding assertion that the projected character is the character of an actual representation of the finite-dimensional Langlands dual Lie algebra. The source attributes that finite-dimensional assertion to McGerty; the affine version is presented as a conjecture.