Self-conjugate CMPs of odd powers for odd m

Determine whether any self-conjugate conjugate $(m,m)$-ary partitions exist for odd powers of an odd integer $m$.

Background

The constructions in Section 5 provide self-conjugate CMPs of many even and odd powers when the base is even, but the authors observe that, for odd $m$, all currently known families—including the square constructions—produce only even powers of $m$.

Thus the existence of a self-conjugate CMP for an odd power of an odd base remains unresolved. The question concerns existence in the entire class, rather than the particular operator families constructed in the paper.