Finite-interval quasi-classical decomposition conjecture for fermionic sums

Let $C$ be the matrix defining the fermionic sums, let $m$ be a multiplicity vector, let $r\leq s$, and let $W_{C,m}$ denote the quadratic weight appearing in the fermionic sum. For a vector $a$ with $0\leq a\leq m$, write $Ca$ for matrix multiplication and $\operatorname{diag}C$ for the vector of diagonal entries of $C$. Finite-interval quasi-classical decomposition conjecture. The finite-interval fermionic sum satisfies

$$I_{C,m}(q,z\mid r,s)=\left(z^mq^{W_{C,m}}\right)^r\sum_{0\leq a\leq m}\left(z^aq^{W_{C,a}}\right)^{s-r}I_{C,m-a}(q,q^{Ca}z)I_{C,a}(q,z^{-1}q^{-Ca+\operatorname{diag}C}).$$

This is the explicit form of the general quasi-classical decomposition. The source presents it as a conjecture and later proves the conjecture in some cases, so the displayed identity remains open in the stated generality.