Interval conjecture for multiplication by a rectangular $K$-$k$-Schur function

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 $4\le4$ be the strong order on $4\mathcal{P}_k4$, defined by inclusion of the corresponding cores. For $4\lambda\in\mathcal{P}_k4$ and $1\le t\le k$, consider the product $g^{(k)}_{R_t\cup\lambda}$.

Interval conjecture. For every $4\lambda\in\mathcal{P}_k4$ and $1\le t\le k$, there exists $4\mu\in\mathcal{P}_k4$ such that

$$g^{(k)}_{R_t\cup\lambda}=g^{(k)}_{R_t}\sum_{\mu\le\nu\le\lambda}g^{(k)}_\nu.$$

In the case of a single rectangle, the coefficients are observed to be either $0$ or $1$, and the conjecture asserts that their support is an interval in the strong order. It remains open whether this interval description holds for all $k$-bounded partitions and all $1\le t\le k$.