Fermionic formula for restricted one-dimensional sums

Let $n$ and $l$ be positive integers, and let $X^{(l)}_{ u}(\rho)$, $X^{(l)'}_{ u}(\rho)$, $F^{(l)}_{ u}(q)$, and $F^{(l)'}_{ u}(q)$ denote the restricted one-dimensional sums and fermionic expressions defined above. For a partition $u$, write $| u|$ for its size and $u_1$ for its largest part. Fermionic formula conjecture. For a partition $u$ such that $| u|\equiv 0\pmod n$ and $u_1\le l$, and for a partition $\rho$ such that $\rho_1\le n-1$ and $|\rho|\equiv0\pmod n$, one has

$$X^{(l)}_{\nu}(l\Lambda_0)=F^{(l)}_{\nu}(q),\qquad X^{(l)'}_{\rho}(l\Lambda_0)=F^{(l)'}_{\rho}(q).$$

These identities assert fermionic expressions for the restricted one-dimensional sums in the vacuum-module case; the source subsequently uses the first identity as an assumption in deriving further results.