Single-root Fourier-eigenfunction convergence conjecture

For $\varepsilon\in\{0,1\}$, let

$$g^\varepsilon_{n,k}(x)=r^\varepsilon_{n,k}(|x|^2)e^{-\pi|x|^2},$$

where $r^\varepsilon_{n,k}$ has degree at most $2k+\varepsilon$, is nonzero, vanishes at $2,4,\ldots,2k$, and is a linear combination of the relevant radial Fourier-eigenpolynomials, so that $\widehat{g}^\varepsilon_{n,k}=(-1)^\varepsilon g^\varepsilon_{n,k}$. Single-root Fourier-eigenfunction convergence conjecture. With $n$ and $\varepsilon$ fixed, as $k\to\infty$, suitably normalized $g^\varepsilon_{n,k}$ converges to a nonzero Fourier eigenfunction $g^\varepsilon_n$ vanishing at every radius $\sqrt{2j}$, and the convergence is uniform on compact subsets of $\mathbb C$ when the functions are viewed as entire functions of $|x|$. Uniform convergence would imply that the limiting eigenfunctions are entire; the source states the convergence conjecturally and gives no general proof.