Matrix case: Are centered orthogonally invariant matrices with independent entries isotropic?

Determine whether, in the matrix case p=2 and dimension N≥2, a centered real symmetric matrix H with independent entries (up to symmetries) and an orthogonally invariant law is necessarily isotropic; specifically, whether H/||H||_F is uniformly distributed on the unit Frobenius sphere of S^{(2)}(N).

Background

In the p=2 case, orthogonal invariance with independent entries forces off-diagonal entries to be centered and diagonal entries to be identically distributed, so a natural centering yields a matrix that is centered, orthogonally invariant, and still has independent entries. If these conditions implied isotropy, then Letac’s characterization would apply to matrices, mirroring the vector case.

Resolving this would clarify whether the Letac-type route suffices to re-derive the Maxwell/Rosenzweig–Porter characterization in the matrix setting based solely on independence and orthogonal invariance.