K-crystals on rectangular shapes conjecture

Let $\operatorname{SV}^n(k\Lambda_i)$ denote the set of semistandard set-valued tableaux of rectangular shape $k\Lambda_i$, where the rectangle has height or width determined by $\Lambda_i$, and let (H.2) be the stated crystal condition. Rectangular K-crystal conjecture. There exists a K-crystal structure also satisfying (H.2) for $\operatorname{SV}^n(k\Lambda_i)$, that is, for rectangle shapes. Such a structure would extend the paper's K-crystal constructions from rows and columns to rectangles; the source presents this as a possibility and does not establish it.