The pointedness conjecture for highway-path homology classes

Let $S$ be an orientable surface and let $u,v$ be boundary vertices. Fix a basepath $p$ from $u$ to $v$, and let $H(u,v)$ be the set of homology classes [q\boldsymbol{\u}\cup p^*]\in H_1(S,\mathbb{Z}), where $q$ ranges over all highway paths from $u$ to $v$. A subset of $\mathbb{Z}^n$ is called a polyhedron if it is the intersection of $\mathbb{Z}^n$ with a polyhedron in $\mathbb{R}^n$, and a polyhedron is pointed if it contains no line. Pointedness conjecture. The set $H(u,v)$ is a pointed polyhedron. This strengthens the known containment of $H(u,v)$ in the polyhedron defined by the affine inequalities $\langle q,r\rangle\geq\langle p,r\rangle$ for each snake path or cycle $r$; proving pointedness would clarify the structure underlying the associated generating functions.