The $k$-distance set bound and equality conjecture

Let $S$ be a $k$-distance set in an $n$-dimensional normed space, meaning that the set of distances between distinct points of $S$ contains at most $k$ numbers. $k$-distance set conjecture. The size of $S$ is at most

$$|S|\leq (k+1)^n.$$

If equality holds for some $k\geq 1$, then the space must be isometric to $\ell_\infty^n$. This conjecture would generalize Petty's theorem on equilateral sets. The source presents it as open and gives no stronger partial result in the supplied context.