The sweep map bijectivity conjecture for rational Dyck paths

Given coprime positive integers $m$ and $n$, let $\mathcal D_{m,n}$ be the set of rational $(m,n)$-Dyck paths, and let $\Phi$ be the sweep map that sorts the steps of a path according to the ranks of their starting points, where the rank of $(a,b)$ is $bm-an$. Sweep map bijectivity conjecture. The sweep map

$$\Phi:\mathcal D_{m,n}\longrightarrow\mathcal D_{m,n}$$

is a bijective transformation. The conjecture is known in the Fuss case $m=kn\pm1$, but remains open for general coprime pairs $(m,n)$.