Existence of asymptotic densities for mutually abelian-bordered and mutually abelian-unbordered pairs

Determine whether the limits of the proportions of mutually abelian-bordered pairs and mutually abelian-unbordered pairs of binary words of equal length exist, specifically whether \(\lim_{n\to\infty}\mathcal{M}(n)/2^{2n}\) and \(\lim_{n\to\infty}\overline{\mathcal{M}(n)}/2^{2n}\) exist.

Background

The paper defines $\mathcal{M}(n)$ as the number of mutually abelian-bordered pairs $(u,v)$ of binary words with $|u|=|v|=n$, and $\overline{\mathcal{M}(n)}$ as the number of mutually abelian-unbordered pairs of the same length. It derives formulas for these quantities by separating disjoint and overlapping border configurations and by using lattice-path enumeration.

The total number of ordered pairs of binary words of length $n$ is $2^{2n}$. The authors therefore ask whether the normalized counts of mutually abelian-bordered and mutually abelian-unbordered pairs converge as $n$ tends to infinity, which would characterize their limiting densities among all equal-length binary word pairs.