Transversals in quasirandom latin squares

A transversal in an $n\times n$ latin square is a collection of $n$ entries not repeating any row, column, or symbol. Kwan showed that almost every $n\times n$ latin square has $((1+o(1))n\mathrm{/}{e}^2{)}^n$ transversals as $n\to \mathrm{\infty }$. Using a loose variant of the circle method we sharpen this to $(e^{-1\mathrm{/}2}+o(1))n{!}^2\mathrm{/}{n}^n$. Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.