Conjugate-normal matrices with conjugate-normal submatrices

In studying the reduction of a complex $n\times n$ matrix $A$ to its Hessenberg form by the Arnoldi algorithm, T. Huckle discovered that an irreducible Hessenberg normal matrix with a normal leading principal $m\times m$ submatrix, where $1<m<n$, actually is tridiagonal. We prove a similar assertion for the conjugate-normal matrices, which play the same role in the theory of unitary congruences as the conventional normal matrices in the theory of unitary similarities. This fact is stated as a purely matrix-theoretic theorem, without any reference to Arnoldi-like algorithms.