The inhomogeneous X=M fermionic formula conjecture

Let $B_0$ be a tensor product of mutually distinct crystals selected from $B^{r_1,s_1},\ldots,B^{r_L,s_L}$, and let $B_0\otimes B^{r_1,s_1}\otimes\cdots\otimes B^{r_L,s_L}$ be equipped with the inhomogeneous energy functions and combinatorial $R$-matrices described in the source. Let $b^\natural_0=b_0\in B_0$, and let $b^\natural_j\in B^{r_j,s_j}$ be the highest weight element for $1\le j\le L$. Write $k=\operatorname{lev}B_0$.

Inhomogeneous X=M conjecture. There exists $b_0\in B_0$ with $\varphi(b_0)=k\Lambda_0$ such that

$$X(B^{r_1,s_1}\otimes\cdots\otimes B^{r_L,s_L},\lambda,q) =q^cM(W^{(r_1)}_{s_1}\otimes\cdots\otimes W^{(r_L)}_{s_L},\lambda,q),$$

where

$$c=\sum_{0\le i<j\le L}H\bigl(b^\natural_i\otimes (b^\natural_j)^{(i+1)}\bigr).$$

Here $M$ is the fermionic form defined in the source. This extends the homogeneous X=M conjecture to inhomogeneous tensor products; the source notes that it is proved for $X_n=A_n$, but gives no general resolution.