The rationality conjecture for generating functions of surface-network measurements

Let $S$ be an orientable surface, let $u,v$ be boundary vertices, and let $M_{u,v}$ denote the generating function associated with the highway paths from $u$ to $v$, expressed using a chosen basis of $H_1(S)$. A Laurent monomial is a monomial whose exponents may be negative. Rationality conjecture. For an appropriate choice of basis of $H_1(S)$, the generating function $M_{u,v}$ is the product of a Laurent monomial and a rational function in the variables $t_i$. This conjecture sharpens the paper's preceding rationality proposition and asserts a particularly controlled form for the generating function of the boundary measurement.