The linear-programming and sign uncertainty constants conjecture

Let $d\geqslant 1$ and let $\mathbb{A}_{LP}(d)$ and $\mathbb{A}_-(d)$ denote, respectively, the linear-programming and negative sign uncertainty constants defined in the paper.

Linear-programming and sign uncertainty conjecture.

$$\mathbb{A}_{LP}(d)=\mathbb{A}_-(d).$$

This would establish a precise link between the sign uncertainty principle and the sphere-packing problem. It is known in dimensions $1$, $8$, and $24$, while the source presents it as an open conjecture in general.