The minimal critical-set bounds for powers of two

Let $L(2^n)$ be the Latin square of order $2^n$, with $n\geq 4$, and let $C$ be a critical set of minimal size in $L(2^n)$. Writing $S$ for this critical set, so that $|S|=|C|$, the minimal critical-set bound conjecture.

$$121\cdot 4^{n-4}\leq |S|\leq 5^{n-1}.$$

These bounds are motivated by computational results for $L(16)$, $L(32)$, and $L(64)$ and by the difficulty of determining optimal critical sets and unique completions. The source does not provide a resolution of the conjectured bound.