The $l^p$ restriction conjecture for uniformly bounded orthonormal systems

Let $2\le q<p<\infty$, and let $\Phi$ be a finite, uniformly bounded orthonormal system. For a finite collection $\Theta$ of functions, define $K_p^*(\Theta)$ to be the smallest constant such that

$$\left\|\sum_{n=1}^{N}a_n\varphi_n\right\|_p\le K_p^*(\Theta)N^{\frac12-\frac1p}\left(\sum_{n=1}^{N}|a_n|^p\right)^{1/p}$$

for every choice of coefficients, where $\Theta=\{\varphi_1,\ldots,\varphi_N\}$. The $l^p$ restriction conjecture. There exists a subset $\Psi\subset\Phi$ satisfying

$$K_p^*(\Psi)\lesssim 1\quad\text{and}\quad |\Psi|\sim K_q^*(\Phi)^{-2q/p}|\Phi|^{q/p}.$$

This conjecture is an $l^p$ analogue of the corresponding selection result for $K_p$ constants. It would provide large subsets with uniformly bounded $K_p^*$ constant in terms of the $K_q^*$ behaviour of the original orthonormal system; the statement is presented as a conjecture because the proof of the analogous theorem has critical steps that do not directly translate to the $l^p$ setting.