Deguchi's highest-weight-vector conjecture for irreducible quotient representations

Let $\boldsymbol{\hat a}$ be the sequence of highest weight parameters, let $\Omega$ be a highest weight vector, and let $w_A(\boldsymbol{\hat a})$ be the operators defined from the subsequences $A$ used in the algorithm. For sequences $\boldsymbol{k}(\boldsymbol{\ell})$, define

$$\omega_{\Sigma(\boldsymbol{k}(\boldsymbol{\ell}))}=\prod_{j=1}^{s}\left(w_{j^{k_j(1)}}(\boldsymbol{\hat a})\cdots w_{j^{k_j(\ell_j)}}(\boldsymbol{\hat a})\right)\Omega.$$

Deguchi's highest-weight-vector conjecture. Every irreducible quotient of a submodule in $U\Omega$ has a highest weight vector of the form $\omega_{\Sigma(\boldsymbol{k}(\boldsymbol{\ell}))}$. The source says this form is verified in examples, but does not prove the assertion in general.