Global optimality of OFDM over the almost-sure set

Let $N$ be the number of subcarriers, let $\mathbb{U}(N)$ denote the unitary group, and let $\Omega_N$ be the almost-sure set appearing in the assumptions of the relevant asymptotic and local optimality theorems. For a unitary waveform basis $\mathbf{U}\in\mathbb{U}(N)$, let $f(\mathbf{U})$ denote the ranging Cramér–Rao bound, and let $\mathbf{F}_N^H$ represent the OFDM basis.

Global optimality of OFDM over the almost-sure set. Under the same assumptions as the preceding asymptotic and local optimality theorems, OFDM is the global minimizer of the ranging CRB over $\Omega_N$; that is, for every $\mathbf{U}\in\mathbb{U}(N)$,

$$\mathbb{E}\!\left[f(\mathbf{U})\middle|\Omega_N\right]\geq \mathbb{E}\!\left[f(\mathbf{F}_N^{H})\middle|\Omega_N\right].$$

The conjecture would close the narrow gap left by the paper's frequency-spread and local optimality results, which exclude broad classes of non-OFDM bases and small perturbations of OFDM but do not cover every unitary waveform basis. Its status is not established by the supplied text.