Knaster's problem for points on a sphere

Let $S^{d-1}$ be a unit sphere in $\mathbb R^d$. Suppose we are given $d$ points $x_1, \ldots, x_d\in S^{d-1}$ and a continuous function $f: S^{d-1}\to \mathbb R$. Knaster's conjecture. There exists a rotation $\rho\in SO(d)$ such that

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

The conjecture is known to hold for $d=3$, but counterexamples are known for certain functions and point sets when $d=61$ and for all $d\ge 67$; its general form is therefore refuted.