Full runner removal conjecture for Ariki–Koike algebras

Let $\bm\lambda,\bm\mu\in\mathcal{P}^r$ lie in a block $B$ of $\mathcal{H}_{r,n}$, with $\bm\mu$ $e$-multiregular. Obtain $\bm\lambda^{+\varnothing}$ and $\bm\mu^{+\varnothing}$ from the $e$-abacus displays of $\bm\lambda$ and $\bm\mu$ by adding an empty runner in each component. Full runner removal conjecture. The $v$-decomposition numbers satisfy

$$d_{\bm\lambda\bm\mu}^e(v)=d_{\bm\lambda^{+\varnothing}\bm\mu^{+\varnothing}}^{e+1}(v).$$

This is motivated by the preceding example, where the equality holds for the displayed multipartitions; the supplied text gives no resolution status.