Forest number of Cartesian products of paths versus trees

Determine whether, for all trees T and T' of respective orders n and n', the forest number satisfies f(P_n □ P_{n'}) ≤ f(T □ T'), where P_n and P_{n'} are paths and □ denotes the Cartesian product.

Background

The paper resolves the opposite extremal inequality, proving that among Cartesian products of two trees of fixed respective orders, the product of two stars has maximum forest number: f(T □ T') ≤ f(S_n □ S_{n'}). The concluding remarks identify the reverse comparison with the Cartesian product of paths as a separate conjecture originating in Wang and Wu [38, Conjecture 5.1].

The problem remains unresolved in part because the exact decycling number, and equivalently the forest number, of the Cartesian product P_n □ P_{n'} is not known for all pairs of orders. Previous work established partial results for Cartesian products of paths, but no general solution is known.