Conjecture on nonnegative coefficients in the highest-weight inner-product recursion

Let $|\lambda\rangle=E_{j_{n(w)}}^-\cdots E_{j_1}^-|\Lambda_s\rangle$ be a state in a finite-dimensional irreducible representation, and let $n_k$ and $\Lambda_i^k$ denote the coefficients and highest-weight data appearing on the right-hand side of the recursion identity for its inner product. Nonnegativity conjecture. The coefficients satisfy

$$n_k(\Lambda^k_i-(n_k-1))\geq 0.$$

This conjecture concerns the positivity of the terms governing norms in the proposed recursion algorithm and is intended to support the study of unitarity in finite-dimensional irreducible representations. The supplied text does not state whether the conjecture has been resolved.