Cofiber Morava K-theory dimension formula conjecture for real Grassmannians

Let $C_d({\mathbb R}^m)$ be the cofiber of the inclusion $\operatorname{Gr}_d({\mathbb R}^{m-1})\hookrightarrow\operatorname{Gr}_d({\mathbb R}^m)$, and let $\bar k_n(C_d({\mathbb R}^m))$ denote its reduced Morava K-theory dimension. Let $n\geq 1$, let $\epsilon$ be $0$ or $1$, and let $l\geq 0$, with $m=2^{n+1}-\epsilon+2l$. Cofiber Morava K-theory dimension formula conjecture. Equality should hold in the proved lower bound, namely

$$\bar k_n(C_d({\mathbb R}^m))=\sum_i\binom{2^{n+1}-1-\epsilon}{d-1-2i}\binom{l}{i}.$$

The conjecture is motivated by a corresponding calculation of the $Q_n$-homology of the cofiber and would determine the lower bound exactly. It remains open.