Profile-count conjecture for rigid indecomposable rank 2 modules

Let $N_{k,n}$ denote the quantity defined in the paper for the relevant profile count, and consider rank $2$ modules in ${\rm CM}(B_{k,n})$.

Profile-count conjecture. The number of profiles of rigid indecomposable rank $2$ modules in ${\rm CM}(B_{k,n})$ is $N_{k,n}$.

The statement appears as a conjectural counting assertion in the paper, relating rigid indecomposable rank $2$ modules to the combinatorial quantity $N_{k,n}$. Its general status is not resolved in the supplied text.