Generic dilation conjecture for the discrete fractal uncertainty principle

Fix an integer base $M$ and alphabet $\mathcal A$ with $0<\delta<1$, let $\mathcal C_k$ be the associated discrete Cantor set, and put $N=M^k$. For $\alpha\in[1,M]$, define the dilated Fourier transform by

$$\mathcal F_{N,\alpha}u(j)=\frac1{\sqrt N}\sum_{\ell=0}^{N-1}\exp\left(-\frac{2\pi i\alpha j\ell}{N}\right)u(\ell).$$

Generic dilation conjecture. There exists $\beta>\max(0,\frac12-\delta)$ depending only on $\delta$ such that, for a generic choice of $\alpha\in[1,M]$,

$$\|\mathbf 1_{\mathcal C_k}\mathcal F_{N,\alpha}\mathbf 1_{\mathcal C_k}\|_{\mathbb C^N\to\mathbb C^N}=\mathcal O(N^{-\beta})\quad\text{as }k\to\infty.$$

A generic dilation is conjectured to improve the elementary discrete FUP exponent uniformly in the base and alphabet, whereas the undilated exponent can be arbitrarily close to the elementary bound.