Common reducibility conjecture for Kostka pairs

Let $(\lambda,\mu)\in {\sf Kostka}_{r}^{\mathbb Z}$, and call a decomposition

$$(\lambda,\mu)=(\lambda^{\bullet},\mu^{\bullet})+(\lambda^{\circ},\mu^{\circ})$$

commonly reducible when $\lambda^{\bullet}$ and $\mu^{\bullet}$ are obtained by choosing common columns of $\lambda$ and $\mu$. Common reducibility conjecture. If $\lambda_1>r$, then $(\lambda,\mu)$ is commonly reducible. The source describes this as a strengthening of an earlier theorem and gives no resolution.