Uniqueness of partitions of powers of 13 and 7

Determine whether powers of 13 and powers of 7 have unique conjugate $(13,7)$-ary partitions.

Background

For the relatively prime pair (m,q)=(13,7)(m,q)=(13,7)), the modular restrictions developed in the paper do not force uniqueness of partitions of powers of either base. The only known examples are the trivial rectangular partitions generated by MkM^k or RkR^k.

The authors state that computational searches have found no counterexamples, leaving unresolved whether non-rectangular conjugate (13,7)(13,7)-ary partitions of powers of 13 or 7 exist.

References

Similarly, for $(13,7)$ it is unknown whether the powers of 13 and 7 are uniquely partitioned or not.

— A study of $m$-ary partitions whose conjugates are $q$-ary  (2609.08799 - Dietz et al., 8 Sep 2026) in Section 4, Example following Theorem 4.?, discussion of the pairs $(5,3)$ and $(13,7)$