The homogeneous X=M fermionic formula conjecture

Let $B=B^{r,s}$, set $k=\left\lceil\frac{s}{t_r}\right\rceil$, and choose $b_0,b^\natural\in B$ such that

$$\varphi(b_0)=k\Lambda_0,\qquad \operatorname{wt}b^\natural=s\overline{\Lambda}_r.$$

Let $X(B^{\otimes L},\lambda,q)$ be the classically restricted one-dimensional sum, let $M(W^{(r)\otimes L}_s,\lambda,q)$ be the corresponding fermionic form, and let $H$ be the energy function.

X=M conjecture. One has

$$X(B^{\otimes L},\lambda,q)=q^cM(W^{(r)\otimes L}_s,\lambda,q),$$

where

$$c=\frac{L(L-1)}{2}H(b^\natural\otimes b^\natural)+LH(b_0\otimes b^\natural).$$

Here $M$ is the fermionic form defined in the source. This conjecture proposes equality between the crystal one-dimensional sum and the fermionic formula, up to the stated energy normalization; the source gives no resolution status.