The lower-bound conjecture for equilateral sets in normed spaces

Let $X$ be an $n$-dimensional normed space, and let $e(X)$ denote the maximum cardinality of an equilateral subset of $X$. Lower-bound conjecture. If $\dim X=n$, then

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

This is the natural lower-bound analogue of the Euclidean simplex construction and would determine the correct universal lower bound in every finite dimension. However, for each $n\geq 4$ this is open; the source notes that the best known result is the theorem of Brass and Dekster giving $e(X)\geq c(\log n)^{1/3}$ for an absolute constant $c$.