Is energy diffusion bounded above by the butterfly velocity

Blake (2016) found $D = C v_B^2/(2 \pi k_B T/\hbar)$ with $C$ of order one in holographic theories, where $v_B$ is the butterfly velocity of the OTOC front; its charge-diffusion form fails in inhomogeneous holographic models (Lucas and Steinberg 2016). Hartman, Hartnoll and Mahajan (2017) conjectured an upper bound $D <\sim v^2 \tau_{\mathrm{eq}}$ ($v$ a characteristic velocity, $\tau_{\mathrm{eq}}$ the local equilibration time), and in inhomogeneous SYK chains energy diffusion obeys $D \le v_B^2/\lambda_L$ (Gu, Lucas, Qi 2017). For local chaotic lattice systems with a well-defined Lyapunov regime, does $D_{\mathrm{energy}} \le C v_B^2/\lambda_L$ hold with a universal $C$ of order one? Answer yes or no, with a counterexample or derivation.

A general derivation from hydrodynamics and operator growth, or an explicit lattice model with $D_{\mathrm{energy}}\,\lambda_{\mathrm{L}}/v_{\mathrm{B}}^2 \to \infty$.