Asymptotic equality conjecture for positive and negative Fourier eigenfunctions

For $s\in\{+1,-1\}$ and dimension $d$, let $\mathrm{A}_s(d)$ denote the corresponding optimal radius in the Fourier-eigenfunction uncertainty principle. Asymptotic equality conjecture. Both limits

$$\lim_{d\to\infty}\frac{\mathrm{A}_+(d)}{\sqrt d} \qquad\text{and}\qquad \lim_{d\to\infty}\frac{\mathrm{A}_-(d)}{\sqrt d}$$

exist and are equal. The paper gives upper and lower bounds of the correct order for both quantities, but no proof of existence or equality of the limits is known.