Paley ETF RIP-constant conjecture

Let $p\equiv 1\pmod{4}$ be prime, let $\Phi_p$ be the Paley equiangular tight frame, and let $\delta_K=\delta_K(\Phi_p)$ denote its RIP constant of order $K$. Paley ETF RIP-constant conjecture. For every $K\geq 1$,

$$\delta_K=O\left(\sqrt{\frac{K}{p}}\,\log K\,\log p\right).$$

Consequently, for constants $c_1,c_2>0$ and $0<\delta<\sqrt{2}-1$, $\Phi_p$ has the $(c_1p/\log^{c_2}p,\delta)$-RIP. The conjecture was proposed as a random-model prediction for Paley ETFs, remains open, and is described as difficult; it is consistent with the conjectured polylogarithmic clique number of Paley graphs.