Equidistribution conjecture for squared sextic Gauss sums

Let $\Psi_6(X)$ be the primitive Dirichlet characters of order $6$ with conductor at most $X$, and let $\tau(\chi)$ be the Gauss sum of $\chi$. Sextic Gauss-sum equidistribution conjecture. The points

$$\frac{\tau(\chi)^2}{q},\qquad \chi\in\Psi_6(X),\quad q=\operatorname{cond}(\chi),$$

should equidistribute on the unit circle as $X\to\infty$. This extends the established equidistribution result for totally sextic characters to the full sextic family, where the paper identifies a slower expected rate of equidistribution.