The self-dual relaxed lattice equality conjecture

A relaxed lattice is a tempered distribution $g$ such that $g$ and its Fourier transform $\widehat{g}$ have the form

$$\sum_{i \ge 0} a_i \delta_{r_i}$$

where $a_i \ge 0$ for all $i$ (not all zero) and $0=r_0<r_1<r_2<\cdots$. It is self-dual when $\widehat{g}=g$.

Self-dual relaxed lattice conjecture. In every dimension, the largest possible value of $r_1$ among self-dual relaxed lattices equals the smallest value of $r$ possible in Theorem 3.2 of Cohn and Elkies.

This conjecture identifies the optimum of the rescaled self-dual relaxed-lattice problem with the optimum in the earlier dual-program bound for sphere packing. The supplied text does not state whether this equality has been proved or disproved.