Logarithmic degree-boundedness for intersection graphs of proper minor-closed classes

A graph class $\mathcal{F}$ is proper if some graph is not isomorphic to any graph in $\mathcal{F}$. Write $\int(\mathcal{F})$ for the class of intersection graphs of collections of vertex-subsets inducing connected subgraphs of graphs in $\mathcal{F}$. Intersection-graph degree conjecture. For every proper minor-closed class of graphs $\mathcal{F}$, there is a constant $c=c(\mathcal{F})$ such that every graph $G\in \int(\mathcal{F})$ has a vertex of degree at most

$$c\tau(G)\log(\tau(G)).$$

The conjecture proposes a uniform growth rate for degree-bounding functions of these intersection-graph classes. The source notes that degree-boundedness is known, while this stronger quantitative bound remains open.