Exponential smallness of discrete fractal uncertainty exponents

For an integer base $M$ and an alphabet $\mathcal A\subset\{0,\ldots,M-1\}$, let $\delta(M,\mathcal A)=\log_M|\mathcal A|$, let $\mathcal C_k$ be the associated discrete Cantor set, and set $N=M^k$. Define

$$\beta(M,\mathcal A):=-\limsup_{k\to\infty}\frac{\log\|\mathbf 1_{\mathcal C_k}\mathcal F_N\mathbf 1_{\mathcal C_k}\|_{\mathbb C^N\to\mathbb C^N}}{\log N}.$$

Exponential-smallness conjecture. Fix $\delta\in(1/2,1)$. There exists a sequence of pairs $(M_j,\mathcal A_j)$ such that

$$\delta(M_j,\mathcal A_j)\to\delta,\qquad \beta(M_j,\mathcal A_j)\to0.$$

This predicts that for dimensions above $1/2$, the improvement over the elementary exponent may be arbitrarily small; the source presents it as suggested by numerics.