Extension of the crystal commutor realization to quasi-minuscule embeddings

Let $B_{\boldsymbol{\pi}'}$ and $B_{\boldsymbol{\pi}}$ be the crystals embedded as in equation, and let Theorem denote the stated agreement between the combinatorial realization and the commutor of Henriques and Kamnitzer. Extension conjecture. Theorem holds without the restriction on the embeddings of the crystals $B_{\boldsymbol{\pi}'}$ and $B_{\boldsymbol{\pi}}$, i.e., it holds for embeddings into tensor products of minuscule and quasi-minuscule crystals. This conjecture extends the experimentally checked realization from embeddings into tensor products of minuscule crystals to embeddings that may also involve quasi-minuscule crystals; its resolution is not given in the supplied text.