Fermionic formula conjecture for generalized Kostka polynomials

Let $N\geq 1,n\geq 2$ be integers, $\lambda$ a partition with $\operatorname{height}(\lambda)\leq n$, and $\boldsymbol L$ an $n\times N$ matrix with nonnegative integer entries. Let $F(\boldsymbol L,\lambda)$ denote the fermionic representation defined in the paper and $K(\boldsymbol L,\lambda)$ the corresponding generalized Kostka polynomial. Fermionic formula conjecture. Then

$$F(\boldsymbol L,\lambda)=K(\boldsymbol L,\lambda).$$

The theorem is proved in the paper under additional monotonicity restrictions on the entries of $\boldsymbol L$; the conjecture extends it to arbitrary nonnegative entries. The claim concerns the equality between the fermionic and generalized Kostka-polynomial descriptions.