Failure of radial Schwartz interpolation on the density-one hexagonal lattice radii

Let $r_1,r_2,\dots$ be the positive real numbers of the form

$$(4/3)^{1/4}\sqrt{j^2+jk+k^2},$$

where $j$ and $k$ are integers. Radial Schwartz functions $f\colon{\mathbb R}^2\to{\mathbb R}$ are considered through their values and radial derivatives at these radii, together with the corresponding Fourier-transform data. Hexagonal interpolation-failure conjecture. Such radial Schwartz functions are not uniquely determined by the values $f(r_n)$, $f'(r_n)$, $\widehat{f}(r_n)$, and $\widehat{f}\,'(r_n)$ for integers $n\geq 1$. The conjecture is motivated by the relative sparsity of the distinct distances in the density-one hexagonal lattice and by numerical computations; the source gives no proof or resolution.