Cohn–Elkies sphere-packing conjecture in dimensions 2, 8, and 24

Let $n\in\{2,8,24\}$, and let $f$ satisfy the hypotheses of the Cohn–Elkies linear-programming bound. Write $r$ for the radius parameter in that bound. Cohn–Elkies sphere-packing conjecture. There exists such a function $f$ with

$$r=\begin{cases}(4/3)^{1/4}&\text{if }n=2,\\sqrt{2}&\text{if }n=8,\\2&\text{if }n=24. \end{cases}$$

This conjecture asserts that a single auxiliary function attains the conjectured optimal sphere-packing bounds in these exceptional dimensions. It is proved in dimension $8$, while the corresponding claims in dimensions $2$ and $24$ are also known by later work; the supplied source specifically notes the $n=8$ resolution.