Integral non-group-theoretical modular categories of dimension $p^2q^2$

We construct all integral non-group-theoretical modular categories of dimension $p^2q^2$, where $p$ and $q$ are distinct prime numbers, establishing that a necessary and sufficient condition for their existence is that $p \mid q+1$, and their rank is $p^2 + \frac{q^2 - 1}{p}$.