Pseudo-dimension matching limit for dense graph sequences

Let $G_n$ be a sequence of graphs admitting a graphon limit $\mathcal{G}$, with edge weights following a distribution of pseudo-dimension $q$. Suppose that there exists $\varepsilon>0$ such that the measure on $[0,1]^2$ given by $\mathcal{G}$ has density at least $\varepsilon$ everywhere, and let $M(G_n)$ denote the minimum total weight of a perfect matching. Dense graph matching conjecture. The quantity

$$n^{-1+1/q}M(G_n)$$

converges in probability to a constant depending only on $q$ and $\mathcal{G}$. This is presented as a speculation extending the proved convergence result for complete bipartite graphs; the source does not provide a resolution, so the conjecture remains open.