Global EISL optimality conjecture for CP-free OFDM under sub-Gaussian constellations

Let $N$ be the number of signaling dimensions, let $\mathbf{U}\in\mathbb{U}(N)$ denote the unitary signaling basis, and let $\mathbf{F}_N$ be the normalized $N$-point discrete Fourier transform matrix. Consider communication-centric ISAC signals without a cyclic prefix, with random symbols drawn from a sub-Gaussian constellation, and measure ranging performance by the expected integrated sidelobe level (EISL) of the aperiodic autocorrelation function. CP-free OFDM global-optimality conjecture. OFDM is the global minimizer of the EISL among all such signals. Equivalently, $\mathbf{U}^\star=\mathbf{F}_N^H$ is the globally optimal solution to the associated $\ell_4$-norm maximization problem. The paper proves only local optimality of OFDM for this problem; numerical results suggest that no other CP-free waveform has a lower sidelobe level, but global optimality remains open.