Polynomial bound for strong coloring numbers by expansion

Establish whether there exists a polynomial p(x,y) such that, for every graph G and every positive integer r, the strong r-coloring number satisfies col_r(G) at most p(r,nabla_r(G)).

Background

The paper gives polynomial bounds for r-admissibility in terms of topological expansion and derives corresponding bounds for weak coloring numbers. It asks whether an analogous polynomial relationship holds between the strong coloring number and the ordinary expansion parameter.