Counting unequal-length mutually abelian-bordered binary pairs

Determine the number of mutually abelian-bordered pairs of binary words \((u,v)\) when the word lengths are unequal, that is, when \(|u|\ne |v|\).

Background

The main enumeration in the paper concerns mutually abelian-bordered pairs of binary words of equal length, and the notation $\mathcal{M}(n)$ is introduced specifically for pairs satisfying $|u|=|v|=n$. The abelian-border conditions require matched factors to have equal lengths, but the paper does not extend its enumeration to pairs whose total word lengths differ.

The conclusion explicitly identifies the unequal-length case as a further counting problem, asking for an enumeration beyond the equal-length setting treated in the paper.