Is there a small skew Cayley transform with zero diagonal?

The eigenvectors of an Hermitian matrix H are the columns of some complex unitary matrix Q. For any diagonal unitary matrix Ω the columns of Q · Ω are eigenvectors too. Among all such Q · Ω at least one has a skew-Hermitian Cayley transform S ≔ (I + Q · Ω)−1 · (I − Q · Ω) with just zeros on its diagonal. Why? The proof is unobvious, as is the further observation that Ω may also be so chosen that no element of this S need exceed 1 in magnitude. Thus, plausible constraints, easy to satisfy by perturbations of complex eigenvectors when Hermitian matrix H is perturbed infinitesimally, can be satisfied for discrete perturbations too. But if H is real symmetric, Q real orthogonal and Ω restricted to diagonals of ±1’s, then whether at least one real skew-symmetric S must have no element bigger than 1 in magnitude is not known yet.