A sufficient condition for an ergodic system to be weak mixing

Let $mathbf X := (X, mathcal S, mu, T)$ be an ergodic measure preserving system with finite measure such that for every increasing sequence ${n_k}$ of natural numbers with positive lower density, we have

$$frac{1}{N} sum_{i=0}^{N-1} f(T^{n_i} x) to int_X f dmu$$

as $n to infty$.

Question: Is $mathbf X$ necessarily weak mixing?