Franks–Misiurewicz conjecture for rotation sets of torus homeomorphisms

Let $f\colon\mathbb{T}^2\to\mathbb{T}^2$ be a homeomorphism homotopic to the identity and let $\tilde f\colon\mathbb{R}^2\to\mathbb{R}^2$ be a lift of $f$. Suppose that the rotation set $\rho(\tilde f)$ is a non-degenerate line segment. Franks–Misiurewicz conjecture. The following dichotomy holds: either $\rho(\tilde f)$ has irrational slope and one of its extreme points belongs to $\mathbb{Q}^2$, or $\rho(\tilde f)$ has rational slope and contains infinitely many rational points. The second case remains open according to the source, namely whether there exists a homeomorphism whose rotation set has rational slope and is disjoint from $\mathbb{Q}^2$; this is one of the paper's main motivations.