Makeev's four-point conjecture on the sphere

Let $S^2$ be a unit sphere in $\mathbb R^3$. Suppose we are given $4$ points $x_1, \ldots, x_4\in S^2$ lying on a single circle, and a continuous function $f:S^2\to\mathbb R$. Makeev's spherical four-point conjecture. There exists a rotation $\rho\in SO(3)$ such that

$$f(\rho(x_1)) = f(\rho(x_2)) = f(\rho(x_3)) = f(\rho(x_4)).$$

The conjecture is known for some particular classes of functions, but the source gives a counterexample and states that it is false in general.