Exponential decay conjecture for the robustness parameter of minimally redundant frames

Let $\mathcal{F}=\{f_j\}_{j=1}^{2n-1}$ be a frame with $\|f_j\|\leq L$ for all $f_j\in\mathcal{F}$, and let $\omega$ denote the robustness parameter discussed above. Exponential decay conjecture. There exist constants $C>0$ and $0<\beta<1$, independent of $n$, such that

$$\omega\leq CL\beta^{-n}.$$

The claim concerns the decay of the robustness of phaseless reconstruction when the number of frame elements is the minimal redundant number $m=2n-1$. Earlier estimates give only polynomial upper bounds, and the optimal rate remains open.