Longest row-cycle conjecture for random Latin squares

Let $L$ be a Latin square of order $n$, and let $\ell(L)$ denote the length of its longest row cycle. Longest row-cycle conjecture. As $n\to\infty$, asymptotically almost all Latin squares of order $n$ satisfy

$$\ell(L)=n.$$

This conjecture is based on numerical evidence and asserts that a uniformly typical Latin square has a row cycle of maximum possible length. The source gives no resolution, so it remains open.