Numerical evidence suggests that $p_2(n) \geq n p(n)$ for large $n$.

Here $p(n)$ is the number of partitions of $n$, and $p_2(n)$ is the number of bipartitions of $n$, i.e., ordered pairs of partitions $(\alpha,\beta)$ with $|\alpha|+|\beta|=n$.  I would be grateful for a proof or a disproof.