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?