The equilateral dimension conjecture for finite-dimensional normed spaces

Let $X$ be an $n$-dimensional normed space. The equilateral dimension conjecture.

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

Here $e(X)$ denotes the maximal cardinality of an equilateral set in $X$, that is, a set of pairwise equidistant points. This conjecture asks for a universal lower bound on the size of equilateral sets in finite-dimensional normed spaces; it is proved for $n\leq 4$ but remains open for all $n\geq 5$.