Rubinstein–Sarnak conjecture on unbiased prime number races

Let $q$ be a positive integer and let $(a_1,\dots,a_r)\in\mathcal{A}_r(q)$, where $\{q;a_1,\dots,a_r\}$ denotes the corresponding race and “unbiased” means that every ordering has density $1/r!$. The condition referred to as assumption (1.3) is

$$a_2\equiv a_1\rho \pmod q,\qquad a_3\equiv a_1\rho^2 \pmod q,$$

for some $\rho\ne 1$ with $\rho^3\equiv 1\pmod q$.

Rubinstein–Sarnak conjecture. When $r\geq 3$, the race $\{q;a_1,\dots,a_r\}$ is unbiased if and only if $r=3$ and the residue classes $a_1,a_2,a_3$ satisfy assumption (1.3).

This conjecture seeks a complete characterization of unbiased races with at least three competitors. Symmetry of the associated density is known to force the stated configuration, but unbiasedness could a priori occur without symmetry.