Petty's conjecture on equilateral sets in normed spaces

Let $(X,\|\cdot\|)$ be a normed space of dimension $n$. A set $S\subseteq X$ is equilateral if there is a constant $c>0$ such that $\|x-y\|=c$ for all distinct $x,y\in S$, and let $e(X)$ denote the largest cardinality of an equilateral set in $X$.

Petty's conjecture. For all normed spaces $X$ of dimension $n$,

$$e(X)\geq n+1.$$

The conjecture gives a general lower bound for equilateral sets in finite-dimensional normed spaces. It is known in dimension at most four, while for $n\geq 5$ it remains open except for some special classes of norms.