Kelly–Trotter product dimension conjecture in general

Determine whether every pair of finite posets P and Q satisfies the lower bound dim(P × Q) ≥ dim P + dim Q − 2, beyond the case dim P = dim Q = 3 established in the paper.

Background

For finite posets P and Q, the Cartesian product always satisfies max{dim P, dim Q} ≤ dim(P × Q) ≤ dim P + dim Q. Kelly and Trotter proposed the stronger lower bound dim(P × Q) ≥ dim P + dim Q − 2 for all finite posets.

The paper proves this conjectured inequality when both factors have dimension three, showing that dim(P × Q) ≥ 4 in that case. The quoted passage indicates that the conjecture remains unresolved in full generality because no counterexample is known.