Kusner's conjecture is false for every $p>4$

False for every $p>4$. This is strictly stronger than the catalog's August entry (arXiv:2608.14013, significance 30), which exhibited $n+2$ equilateral points for a single exponent and only located the infimum of failing exponents in $[4,5)$. Together with Swanepoel for $1<p<2$ and Ge, Xu and Zhou for $2 \le p \le 4$, Kusner's conjecture is now settled for all $1<p<\infty$: true exactly on $[2,4]$.

False for every $p>4$. This is strictly stronger than the catalog's August entry (arXiv:2608.14013, significance 30), which exhibited $n+2$ equilateral points for a single exponent and only located the infimum of failing exponents in $[4,5)$. Together with Swanepoel for $1<p<2$ and Ge, Xu and Zhou for $2 \le p \le 4$, Kusner's conjecture is now settled for all $1<p<\infty$: true exactly on $[2,4]$.