Root-location conjecture for an extremizer in dimension 28

Let $\mathcal{A}_+(28)$ denote the class of radial Schwartz functions on $\mathbb{R}^{28}$ satisfying the positive Fourier-eigenfunction uncertainty conditions. Dimension-28 root-location conjecture. There exists a nonzero radial Schwartz function $g\in\mathcal{A}_+(28)$ such that $\widehat g=g$, $g(0)=0$, and $r(g)=\mathrm{A}_+(28)$, whose nonzero roots have radii

$$\sqrt{2j+o(1)}$$

as $j\to\infty$, beginning with $j=2$. Numerical root locations strongly suggest this structure, but existence of such an extremizer and the asserted root asymptotics are not proved.