Unitary congruence to a conjugate-normal matrix

A matrix $A\in M_n(\mathbb C)$ is said to be conjugate-normal if $AA^*=\overline{A^*A}.$ The following proposition (which is the congruence analog of a recent result of T. G. Gerasimova) is proved: A matrix $B\in M_n(\mathbb C)$ is unitarily congruent to a conjugate-normal matrix $A$ if and only if
$$ \mathrm{tr}[(\bar AA)^i]=\mathrm{tr}[(\bar BB)^i],\qquad i=1,\dots,n, $$
and
$$ \|A\|_F=\|B\|_F. $$
This proposition dramatically reduces the amount of computational work for verifying unitary congruence as compared to the case of general matrices $A$ and $B$.