Polynomial strong-coloring bound with maximum degree

Establish whether there exists a polynomial p(x,y,z) such that every graph G of maximum degree Delta satisfies col_r(G) at most p(r,nabla_r(G),Delta) for every positive integer r.

Background

The preceding problem is already unresolved for general graphs, and the paper observes that the issue appears even for bounded-degree graphs. This formulation asks whether adding the maximum degree as a parameter yields a polynomial bound.