A new lower bound for deterministic pop-stack-sorting

The pop-stack-sorting process is a variation of the stack-sorting process. We consider a deterministic version of this process. We prove a lemma which characterises interior elements of increasing runs after $t$ iterations of the process and provide a new lower bound of $\frac{3}{5}n$ for the number of iterations of the process to fully sort a uniformly randomly chosen permutation of length $n$.