Morava K-theory dimension formula conjecture for real Grassmannians

Let $n\geq 1$, let $\epsilon$ be $0$ or $1$, and let $l\geq 0$. Set $m=2^{n+1}-\epsilon+2l$, and let $k_n(\operatorname{Gr}_d({\mathbb R}^m))$ denote the dimension of $K(n)^*(\operatorname{Gr}_d({\mathbb R}^m))$ over $K(n)_*$. Morava K-theory dimension formula conjecture.

$$k_n(\operatorname{Gr}_d({\mathbb R}^m))=\sum_{i=0}^{\left\lfloor d/2\right\rfloor}\binom{2^{n+1}-\epsilon}{d-2i}\binom{l}{i}.$$

This conjecture would sharpen the proved lower bound for Morava K-theory of real Grassmannians and is motivated by the conjectural calculation of their $Q_n$-homology. It remains open.