Real canonical-profile conjecture in arbitrary rank

Let $M\in {\rm CM}(B_{k,n})$ be rigid indecomposable, let $P_M$ be its profile, and let $\varphi(M)$ be the associated root in $J_{k,n}$. A real canonical profile is a canonical profile whose corresponding root is real.

Real canonical-profile conjecture. Let $M\in {\rm CM}(B_{k,n})$ be rigid indecomposable and assume that $\varphi(M)$ is a real root in $J_{k,n}$. Then $P_M$ is a cyclic permutation of a real canonical profile.

The paper proves the corresponding rank $3$, $k=3$ case and proposes this formulation for arbitrary rank. The general assertion remains open.