The minimal critical-set size of $L(9)$

Let $L(9)$ denote the Latin square of order $9$, and let $C$ be a critical set in $L(9)$. The minimum cardinality of a critical set in $L(9)$ is the $L(9)$ minimal critical-set conjecture.

$$29.$$

The preceding discussion gives computational evidence for lower bounds near this value and notes that proving optimality, or finding a smaller solution, remains unresolved. Thus the conjecture asserts that the minimum is exactly $29$.