The sign-identity conjecture for the theta-series kernel

Let $v_0$ be a null vector and let $N_k,q_k$ be the charge data entering the pairings

$$b^{\,0}_{k\ell}=v_0\cdot(N_kq_{\ell}-N_{\ell}q_k).$$

Assume that all $b^{\,0}_{k\ell}$ are nonzero. Let $\Phi_n$ be the function defined in the source, and let ${\rm Sym}$ denote the stated symmetrization. The sign-identity conjecture. The function

$$F_n(\{\check\gamma_i\},y)={\rm Sym}\left\{y^{\sum_{i<j}\gamma_{ij}}\Phi_n(\{\check\gamma_i\})\right\}$$

has a zero of order $n-1$ at $y=1$. The conjecture is intended to supply the vanishing required in the unrefined limit, but no proof is given for all cases in the source.