Conjecture on optimal linear programming functions for Gaussian energy

Fix a dimension $d\geq 3$, density $\rho=1$, and $\alpha>0$. For the potential $p(r)=e^{-\alpha r^2}$, consider the optimal linear programming bound $\widehat{f}(0)-f(0)$ from the stated linear programming proposition, where $f$ is radial and belongs to the Schwartz space. Gaussian linear-programming conjecture. The optimal bound is achieved by some radial Schwartz function $f$ such that the radii $|x|$ satisfying $f(x)=e^{-\alpha|x|^2}$ are exactly the radii $|y|$ satisfying $\widehat{f}(y)=0$; these radii form a discrete infinite set and do not depend on $\alpha$. The conjecture concerns the sharpness and structure of linear programming bounds for Gaussian energy in dimensions at least three; the source presents it as a variant of conjectures of Cohn and Elkies and Cohn and Kumar, with the relevant general interpolation consequences still unresolved.