Hurlbert–Isaak conjecture on square-shaped de Bruijn tori for even k and n in {3,5,7,9}

Prove or refute the conjecture that, for even alphabet sizes k and pattern size n in {3,5,7,9}, square-shaped de Bruijn tori of type (M, M; n, n)k exist if and only if k is a perfect square.

Background

Known results due to Hurlbert and Isaak establish the existence of square-shaped de Bruijn tori for many parameter ranges, with a characterization linked to parity of n and whether k is a perfect square. However, there are exceptional small odd n where the situation is unresolved.

The cited conjecture posits that the same if-and-only-if condition (existence precisely when k is a perfect square for odd n) should also hold for the remaining exceptional cases n ∈ {3,5,7,9} when k is even. This conjecture remains unproven and is explicitly identified as open in the paper.