Ergodic recurrence and bounded gaps between primes

Let $(X,B_X,\mu ,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in B_X$ with $\mu (A)>0$ and $ϵ>0$. For any $m\ge 1$, there exist infinitely many primes $p_0,p_1,\dots ,p_m$ with $p_0<\cdots <p_m$ such that $$\mu (A\cap T^{-(p_i-1)}A)\ge \mu (A{)}^2-ϵ$$ for each $0\le i\le m$ and $$p_m-p_0<C_m,$$ where $C_m>0$ is a constant only depending on $m$, $A$ and $ϵ$.