Polynomial strong-coloring bounds from expansion

Establish whether there exists a polynomial p such that every graph G and every positive integer r satisfy col_r(G) ≤ p(r,∇_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 r-coloring number and ordinary depth-r expansion.