Edge-contraction conjecture for dosp mutation graphs

Let $\mathbb{S}=S_{g,h}$ be a closed surface notation used in the paper, let $k$ be the rank, and let $\mathbb{E}_{\rm good}$ map to the dosp mutation graph via the map $eqref{eq:Pclustertodosp}$. A vertex is everywhere oneblock when its dosp at every puncture is oneblock. Edge-contraction conjecture. When $\mathbb{S}=S_{g,h}$ and $(k,\mathbb{S})\neq(2,S_{0,3})$, this map is an edge contraction onto the induced subgraph on the vertices which are not everywhere oneblock. The surrounding theorem establishes related edge-contraction properties, while this formulation is presented as a conjecture and its resolution is not given.