Average spectral norm of the random matrix with unit norm rows?

Question

Let $P$ be a distribution of $n \times n$ random matrix whose columns are independent unit norm vectors.

(each rows are drawn uniformly at random from the unit sphere)

Is there any idea about the $\|AB^T\|_2$, where $A, B \sim P$?

I expect it might be $\sqrt{n}$......(it's just my hunch)