Uniqueness of partitions of powers of 5

Determine whether powers of 5 have unique conjugate $(5,3)$-ary partitions.

Background

The paper studies when an integer admits a partition whose parts are powers of $m$ and whose conjugate has parts that are powers of $q$. For $(m,q)=(5,3)$, the authors note that powers of $3$ are not uniquely partitioned for sufficiently large exponents, while the available congruence conditions do not resolve the corresponding question for powers of $5$.

The unresolved issue concerns whether every conjugate $(5,3)$-ary partition of $5^k$ is the rectangular partition generated by $M^k$, or whether non-rectangular alternatives eventually occur. The authors report that computational data has not produced a counterexample, but does not establish uniqueness.