Boutot's theorem says that if $X$ is a variety over a field of characteristic 0 with rational singularities, and if $G$ is a reductive group acting on $X$, then the quotient $X/G$ has rational singularities as well.

Is it known whether an analog of this result is true for log terminal singularities?

Namely suppose $X$ is a variety over a field of characteristic 0 with log terminal singularities and an action of a reductive group $G$. Then must $X/G$ also have log terminal singularities?