Deguchi's vanishing relation conjecture for highest weight vectors

Let $\Omega$ be a highest weight vector, let $m_j$ be the multiplicity associated with the parameter $a_j$, and let $w_{j^k}(\boldsymbol{\hat a})$ be the operators defined in the algorithm. Deguchi's vanishing relation conjecture. For $0\leq n\leq m_j$,

$$\sum_{k=1}^{n} k\,w_{j^{k+1}}(\boldsymbol{\hat a})w_{j^{n+1-k}}(\boldsymbol{\hat a})\Omega=0.$$

These relations are used to determine when certain quotient modules have zero dimension. The source reports verification up to $m_j=5$ and a proof up to $n=3$, while higher cases remain conjectural.