Tight bounds for clustered 3-coloring of strong products of bounded treewidth graphs

We resolve Conjecture 8 of Campbell et al. (2025) by constructing an infinite family of graph pairs (H1(n),H2(n)) with treewidth at most 3 such that every 3-coloring of the strong product H1(n)H2(n) has clustering Ω(|V(H1(n)H2(n))|4/7). This lower bound matches the best known upper bound for such products in the absence of degree constraints, thereby establishing the asymptotic tightness of the exponent 4/7. The construction is based on a symmetrized extremal configuration and employs a two-dimensional application of the Hex Lemma to force a monochromatic grid component of the required size.