Nonnegativity conjecture for factorization coefficients of $K$-$k$-Schur functions

Let $4\mathcal{P}_k4$ be the set of $k$-bounded partitions, let $R_t$ denote the rectangular $k$-bounded partition indexed by $t$, and let $P=R_{t_1}^{a_1}\cup\cdots\cup R_{t_m}^{a_m}$. Define coefficients $a_{P,\lambda,\mu}$ by

$$g^{(k)}_{P\cup\lambda}=g^{(k)}_P\sum_\mu a_{P,\lambda,\mu}g^{(k)}_\mu.$$

Nonnegativity conjecture. For every $4\lambda\in\mathcal{P}_k4$ and every $\mu$, one has $a_{P,\lambda,\mu}\ge 0$.

This conjecture predicts positivity of the coefficients arising when a product indexed by a union of rectangles is factored from a $K$-$k$-Schur function. The cited results establish the factorization for unions of distinct rectangular powers, but do not establish this coefficientwise nonnegativity in general.