The uniform Fourier-dimension conjecture for missing digits measures

Let $p>2$ be a prime number and let $\epsilon\in(0,1)$. Uniform Fourier-dimension conjecture. For every digit set $D$ with

$$\#D\geq p^\epsilon,$$

we have

$$\left|\dim_{l^1}\lambda_{p,D}-\dim_{\mathrm{H}}\lambda_{p,D}\right|=o(1),$$

where the $o(1)$ term is uniform across all such digit sets $D$. Here $\lambda_{p,D}$ denotes the missing digits measure associated with $p$ and $D$. The conjecture is posed as a possibly too optimistic strengthening of the structured-digit estimates in Theorem 3; the supplied text does not resolve it.