Smallest odd power with a self-conjugate CMP for even m

Determine whether $m^{m+1}$ is the smallest odd power of an even integer $m$ that admits a self-conjugate conjugate $(m,m)$-ary partition.

Background

A conjugate (m,m)(m,m)-ary partition is a conjugate mm-ary partition, or CMP. Section 5 constructs several families of self-conjugate CMPs whose underlying integer is a power of mm. For even mm, the family (MR)m(MR)^m produces a self-conjugate CMP of mm+1m^{m+1}, and the authors observe that their constructions produce self-conjugate CMPs for all sufficiently large odd powers.

The unresolved question is whether any smaller odd power of an even base can support a self-conjugate CMP. This asks for a minimality result beyond the explicit construction.

References

When $m$ is even, is $m{m+1}$ the smallest odd power of $m$ that posseses a self-conjugate CMP?

— A study of $m$-ary partitions whose conjugates are $q$-ary  (2609.08799 - Dietz et al., 8 Sep 2026) in Section 5, Question following Discussion 5.?, after the summary of self-conjugate CMP families

For any $m$, does $(MR)m$ always generate the smallest non-square CMP of a power of $m$?

— A study of $m$-ary partitions whose conjugates are $q$-ary  (2609.08799 - Dietz et al., 8 Sep 2026) in Section 5, Question immediately following the question about the smallest odd power for even $m$