Conjecture on the largest critical set in a Latin square

Let $\lcs{n}$ denote the size of the largest critical set in any Latin square of order $n$. Largest-critical-set bound conjecture.

$$\lcs{n} \leq n^2-n^{3/2}.$$

This bound is motivated by the proof of the paper's Theorem 3.1 and is described as analogous to a conjecture of Brankovic, Horak, Miller, and Rosa concerning the size of the largest premature partial Latin square. The paper does not report a resolution.