  Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$
and let $\chi$ be a primitive Dirichlet character of prime power conductor
$\mathfrak{q}=p^{\kappa}$ with $p$ prime and $\kappa>12$. A subconvex bound for
the central values of the Rankin-Selberg $L$-functions $L(s,f\otimes g \otimes
\chi)$ is proved in the depth-aspect $$ L\left(\frac{1}{2},f\otimes g \otimes
\chi\right)\ll_{f,g,\varepsilon} p^{3/4}\mathfrak{q}^{15/16+\varepsilon}. $$
