Compatibility conjecture for ordinary and modular Harish-Chandra series in finite unitary groups

Let $G$ be the finite unitary group and let $X_\mu,X_\nu$ be the unipotent $kG$-modules labelled by partitions $\mu,\nu\in\mathcal{P}_n$, while $Y_\mu,Y_\nu$ are the corresponding unipotent $KG$-modules. Compatibility conjecture. If $X_\mu$ and $X_\nu$ lie in the same weak Harish-Chandra series of $kG$-modules, then $\mu$ and $\nu$ have the same $2$-core; equivalently, $Y_\mu$ and $Y_\nu$ lie in the same Harish-Chandra series of $KG$-modules. Thus the partition arising from weak $\ell$-modular Harish-Chandra series is a refinement of the partition arising from ordinary Harish-Chandra series. This predicts compatibility between the ordinary and modular Harish-Chandra-series decompositions; no resolution is supplied in the source.