Interaction-strength conjecture for Bouw-Möller surfaces with divisible parameters

Let $S_{m,n}$ be the Bouw-Möller surface associated with integers $m,n\geq 2$, and let $\operatorname{Int}(\alpha,\beta)$ denote the intersection number of closed curves $\alpha,\beta$, with $l(\alpha)$ and $l(\beta)$ their lengths. Suppose that $m,n\geq 3$, $(m,n)\neq(3,3)$, and $m$ is a multiple of $n$. Set

$$l_0=\sin\frac{\pi}{m}.$$

Interaction-strength conjecture. For every pair of closed curves $\alpha,\beta$ on $S_{m,n}$,

$$\frac{\operatorname{Int}(\alpha,\beta)}{l(\alpha)l(\beta)}\leq\frac{1}{4l_0^2}.$$

Moreover, equality is achieved by two closed curves $\alpha,\beta$ each made of two sides of length $l_0$ and intersecting once. The conjecture concerns the case excluded from the preceding theorem, where the equality construction with curves intersecting twice does not apply; it would determine the algebraic interaction strength for these Bouw-Möller surfaces when $m$ is a multiple of $n$.