Let $\overline{M}_{0,n}$ be the moduli space of Deligne-Mumford stable pointed rational curves, and let us consider the quotient $\widetilde{M}_{0,n} = \overline{M}_{0,n}/S_n$.

Clearly, there is a dominant rational map $\overline{M}_{0,n}\dashrightarrow \widetilde{M}_{0,n}$, and $\widetilde{M}_{0,n}$ is unirational. If $n=4,5$ then $\widetilde{M}_{0,n}$ is a curve and a surface respectively. Thus, it is rational.

Is it known that $\widetilde{M}_{0,n}$ is rational for any $n\geq 3$? If so, do you know a reference for this?