Symmetry conjecture for the two-factor fermionic sum

Let $\mathfrak g$ be a Kac–Moody algebra, let $\beta$ be a positive-root-lattice element, and let $X^{\lambda_1,\lambda_2}_\beta$ be the bilinear fermionic quantity defined from the corresponding vectors. For ADE type, the source identifies it with $X^{(0,\nu)}_{C,m}(q,z)$ and proves $X^{\lambda_1,\lambda_2}_\beta=X^{\lambda_2,\lambda_1}_\beta$. Two-factor symmetry conjecture. The relation

$$X^{\lambda_1,\lambda_2}_\beta=X^{\lambda_2,\lambda_1}_\beta$$

holds for arbitrary $\mathfrak g$. The claim extends the equality proved in the ADE case to arbitrary Lie type. The excerpt gives no proof for the general case, so its status is open.