The conjecture on Salem sets of prescribed non-integer size

Let $s\in (0,d)$ be a non-integer and let $C$ be a positive constant. For a subset $E\subset \mathbb{F}_{p}^{d}$, write $\Phi(E)$ for its maximum nontrivial Fourier coefficient. Consider subsets satisfying

$$p^{s}/C\leq \# E\leq Cp^{s}.$$

Prescribed-size Salem-set conjecture. One has

$$\min_{E}\frac{\Phi(E)}{\sqrt{\# E}}\longrightarrow\infty\quad\text{as }p\longrightarrow\infty,$$

where the minimum is over all subsets $E\subset\mathbb{F}_{p}^{d}$ with the displayed size bounds. This conjecture asserts that no Salem sets of cardinality comparable to $p^{s}$ exist when $s$ is a non-integer. The preceding results show that random sets are weak Salem sets, while the known Salem examples, the discrete paraboloid and sphere, have size roughly $p^{d-1}$; the conjecture concerns the possibility of Salem sets at intermediate, non-integer-dimensional scales.