Counting mutually abelian-bordered pairs over non-binary alphabets

Determine the number of mutually abelian-bordered pairs of words over alphabets with more than two symbols, extending the binary-word enumeration to non-binary words.

Background

The paper restricts its enumeration to the binary alphabet $\Sigma=\{a,b\}$, using two-dimensional lattice paths to represent binary words and to characterize abelian borders. Although abelian equivalence is defined for general alphabets, the counting results and geometric arguments are developed only for binary words.

The conclusion explicitly asks for the number of mutually abelian-bordered pairs of non-binary words, leaving open the corresponding enumeration over larger alphabets.