Conjecture on the determinant of the resonance-relation system

Let $D$ be the determinant of the system of resonance relations, with parameters $x$ and $\Lambda_2,\dots,\Lambda_n$, and let $[z]$ denote the quantum-number factor used in the paper. Nonvanishing determinant conjecture. One has $D\ne0$ unless $x$ and $\Lambda_2,\dots,\Lambda_n$ satisfy one of the equations

$$[x+k]=0,\qquad k=-{\mathfrak m}+2,\dots,{\mathfrak m},$$ $$[\Lambda_{i+1}+\dots+\Lambda_j-k]=0,\qquad k=2,\dots,2{\mathfrak m}-2,$$

for some $1\le i<j\le n$. The preceding argument proves generic nonvanishing, but the asserted complete description of the possible vanishing locus is presented as a computationally motivated conjecture and remains open.