Universal optimality of the 85-point code in complex dimension five

Let $\mathcal C$ be the code constructed in the paper consisting of $85$ points in $\mathbb C^5$, viewed projectively. A code is universally optimal if it minimizes every admissible completely monotonic potential among codes of the same type and size. Universal-optimality conjecture. The code $\mathcal C$ constructed in this section is universally optimal.

The construction improves the previously known bound of size $320$ for a weighted projective $3$-design, and numerical experiments suggest optimality in several settings. A proof of universal optimality is not supplied and remains open.