Lower bound five for the Poljak–Rödl function

Establish whether there exists an integer n such that the Poljak–Rödl function f(n) satisfies f(n) ≥ 5.

Background

The Poljak–Rödl function is defined by f(n) = min{χ(G × H): χ(G), χ(H) ≥ n}. Known results give f(n) ≥ 4 for n ≥ 4, while counterexamples to Hedetniemi’s conjecture provide upper bounds below n. It is not known whether the lower bound can ever be increased to five.