The weak Knaster conjecture

Let $l$ be a positive integer. For an integer $n$, let $S^{n-1}$ be the unit sphere in $\mathbb R^n$, let $X=\{x_1,\ldots,x_l\}$ be any $l$ points on that sphere, and let $f:S^{n-1}\to\mathbb R$ be continuous.

The weak Knaster conjecture. There exists $n=n(l)$ such that, for every such $X$ and $f$, there is a rotation $\rho\in O(n)$ satisfying

$$f(\rho x_1)=f(\rho x_2)=\dots=f(\rho x_l).$$

The source presents this as the weak form of Knaster's conjecture. It also records that Knaster's original claim $n(l)=l$ has counterexamples, while the weaker existence assertion is the stated conjecture.