Polynomial strong-coloring bounds for bounded-degree graphs

Establish whether there exists a polynomial p such that every graph G of maximum degree Δ and every positive integer r satisfy col_r(G) ≤ p(r,∇_r(G),Δ).

Background

The paper states that the preceding polynomial-expansion question already arises for bounded-degree graphs. This variant explicitly includes the maximum degree as a parameter and asks for a polynomial upper bound on the strong r-coloring number.