Conjecture: Isotropy of centered orthogonally invariant tensors with independent entries

Prove that for an even order p and dimension N≥2, any real symmetric tensor H in S^{(p)}(N) whose entries are independent up to symmetries and whose distribution is orthogonally invariant has its centered version H' := H − E(H_{1, …, 1}) · I^{(p)}_N isotropic; that is, H'/||H'||_F is uniformly distributed on the unit Frobenius sphere of S^{(p)}(N).

Background

Maxwell’s theorem for vectors can be deduced from Letac’s result: independence plus isotropy (uniform distribution on the sphere after normalization) implies Gaussianity. The paper asks whether an analogous route can establish the tensor version: if orthogonal invariance together with independence implies isotropy of a suitable centering of the tensor, then Letac’s characterization would yield the Gaussian Orthogonal Tensor Ensemble.

For even p, adding a multiple of the symmetric tensor identity preserves orthogonal invariance but generally breaks isotropy, motivating the specific centering H' = H − E(H_{1,…,1}) * I^{(p)}_N. Establishing isotropy of H' under the stated assumptions would provide a Letac-type extension for tensors and an alternative proof of the Maxwell-type characterization.