Square-root deficit conjecture for transversals in equi-n-squares

An equi-$n$-square is an $n\times n$ array filled with $n$ symbols, each appearing exactly $n$ times. A transversal is a collection of cells sharing no row, column, or symbol; its size is the number of cells. Square-root deficit conjecture. There exists a constant $C>0$ such that every equi-$n$-square has a transversal with size at least

$$n-C\sqrt{n}.$$

This conjecture proposes the expected order of the optimal universal lower bound after Stein's conjecture was disproved. The paper proves a lower bound of the form $n-n^{1-\varepsilon}$ for some $\varepsilon>0$, while its construction gives examples with no transversal of size $n-(1/(2\sqrt{2})+o(1))\sqrt{n}$; determining the sharp constant remains open.