Rankin--Selberg coefficients in arithmetic progressions modulo prime powers

Let $\varepsilon>0$ be given. For prime power moduli $q=p^k$ with $k\geq 2$ and $p\neq 3$, and assuming the Ramanujan--Petersson conjecture for $\GL_2$ Maass forms, we prove that the Rankin--Selberg coefficients $\{\lambda_f(n)^2\}_{n\geq 1}$ have a level of distribution $\theta=2/5+3/305-\varepsilon$ in arithmetic progressions $n \equiv a \bmod q$.