Bounds for $\mathrm{G}{\mathrm{L}}_2\times \mathrm{G}{\mathrm{L}}_2$ $L$-functions in depth aspect

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$ 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 $L(s,f\otimes g \otimes \chi)$isprovedinthedepth-aspect$$ L\left(\frac{1}{2},f\otimes g \otimes \chi\right)\ll_{f,g,\varepsilon} p^{3/4}\mathfrak{q}^{15/16+\varepsilon}. $$