Scrambling in nearly thermalized states at large central charge

We study $2d$ conformal field theory (CFT) at large central charge $c$ and finite temperature $T$ with heavy operators inserted at spatial infinity. The heavy operators produce a nearly thermalized steady state at an effective temperature $T_{\rm eff}\leq T$. Under some assumptions, we find an effective Schwarzian-like description of these states and, when they exist, their gravity duals. We use this description to compute the Lyapunov exponents for light operators to be $2\pi T_{\rm eff}$, so that scrambling is suppressed by the heavy insertions.