An exponential estimate for the extinction time of the branching random walk on a cube

We prove the exponential estimate \begin{equation*} P { s < \tau < \infty } \leq C e^{-q s}, \quad s \geq 0, \end{equation*} where $C,q>0$ are constants and $\tau$ is the extinction time of the supercritical branching random walk (BRW) on a cube. We cover both the discrete-space and continuous-space BRWs.