The left-cell counting conjecture for type AIII Schur algebras

Let $n=2r+1$, let $\boldsymbol{c}_{\lambda}^{\jmath}$ be the two-sided cell associated with a special partition $\lambda$ of $2d+1$ with at most $n$ parts, and consider semistandard domino tableaux of shape $\lambda$, with a distinguished monomino. The left-cell counting conjecture. The number of left cells in $\boldsymbol{c}_{\lambda}^{\jmath}$ equals the number of semistandard domino tableaux of shape $\lambda$ with all entries in the dominoes at most $r+1$ and the entry in the monomino equal to $1$. This conjecture proposes a tableau-theoretic description of left cells in a two-sided cell of $S^{\jmath}(n,d)$, extending the corresponding relationship between left cells, orbital varieties, and standard domino tableaux; its resolution is not specified in the source.