Approximation-ratio conjecture for randomized binary sequence design

Let $\tilde{\boldsymbol{s}}$ be a binary sequence obtained by randomized projection and binary quantization from $\widehat{\boldsymbol{S}}$, where $\widehat{\boldsymbol{S}}$ is the solution to the semidefinite relaxation. Suppose that $\tilde{\boldsymbol{s}}$ satisfies the inequality constraint

$$\|\mathcal{F}_I\tilde{\boldsymbol{s}}\|_2^2\leq\alpha.$$

Define the approximation ratio by

$$\gamma=\frac{\|\mathcal{F}_M\tilde{\boldsymbol{s}}\|_2^2}{\operatorname{Trace}\left(\mathcal{F}_M^H\mathcal{F}_M\widehat{\mathbf{S}}\right)}.$$

Approximation-ratio conjecture. The ratio satisfies

$$\gamma\geq\frac{\pi}{2}-1.$$

This conjecture is motivated by analogous approximation guarantees for other quadratically constrained quadratic programs and is asserted for feasible randomized projections. The authors report that a theoretical proof is elusive and validate the claim numerically.