Finiteness conjecture for DHS complete bipartite graphs

Let $K_{m,n}$ be the complete bipartite graph with positive integers $m$ and $n$. A mixed graph is DHS if it is determined by its Hermitian adjacency spectrum among mixed graphs. Finiteness conjecture. There are only finitely many integers $m$ and $n$ for which the complete bipartite graph $K_{m,n}$ is DHS. The preceding results show that $K_{n,n}$ is not DHS whenever $n$ is not square-free, but it remains undecided whether infinitely many integers $n$ yield DHS graphs $K_{n,n}$; the conjecture asserts the stronger global finiteness statement for all complete bipartite graphs.