Cellularity conjecture for the mirabolic Hecke algebra

Let $H_n(1,0;q)$ be the Hecke algebra with its cellular basis indexed by standard bitableaux, and let $[?]$ denote the quotient algebra with basis elements $\bar{C}^{\underline{\lambda}}_{\mathfrak{s},\mathfrak{t}}$. Define

$$\Lambda'(n)=\{(\lambda^1,\lambda^2)\in\Lambda(n)\mid \lambda^2\text{ is a single column}\}.$$

Cellularity conjecture. The elements

$$\{\bar{C}^{\underline{\lambda}}_{\mathfrak{s},\mathfrak{t}}\mid\underline{\lambda}\in\Lambda'(n)\}$$

are a cellular basis for $\mathcal{R}_n(q)$, so $\mathcal{R}_n(q)$ is a cellular algebra.

This would identify the quotient's natural images of the Hecke algebra's cellular basis as a cellular basis indexed by the specified bipartitions. The supplied text gives a strategy involving reduction of basis elements indexed outside $\Lambda'(n)$, but does not establish the conjecture.