Power-saving error term for q-orbits of closed geodesics

Let $q$ be the modulus, and consider the equidistribution conclusion for the $q$-orbits of closed geodesics given by the first theorem.

Power-saving error-term conjecture. The conclusion of the first theorem holds with a power-saving error term in terms of $q$.

The question is motivated by Duke's power-saving result for closed geodesics in the real quadratic setting. No resolution is supplied in the source.