Sub-convexity bound for $GL(3) times GL(2)$ $L$-functions: $GL(3)$-spectral aspect

Let $phi$ be a Hecke-Maass cusp form for $SL(3, mathbb{Z})$ with Langlands parameters $({bf t}_{i})_{i=1}^{3}$ satisfying $$|{bf t}_{3} - {bf t}_{2}| leq T^{1-xi -epsilon}, quad , {bf t}_{i} approx T, quad , , i=1,2,3$$ with $1/2 < xi <1$ and any $epsilon>0$. Let $f$ be a holomorphic or Maass Hecke eigenform for $SL(2,mathbb{Z})$. In this article, we prove a sub-convexity bound $$L(phi times f, frac{1}{2}) ll max { T^{frac{3}{2}-frac{xi}{4}+epsilon} , T^{frac{3}{2}-frac{1-2 xi}{4}+epsilon} } $$ for the central values $L(phi times f, frac{1}{2})$ of the Rankin-Selberg $L$-function of $phi$ and $f$, where the implied constants may depend on $f$ and $epsilon$. Conditionally, we also obtain a subconvexity bound for $L(phi times f, frac{1}{2})$ when the spectral parameters of $phi$ are in generic position, that is $${bf t}_{i} - {bf t}_{j} approx T, quad , text{for} , i eq j, quad , {bf t}_{i} approx T , , , i=1,2,3.$$