Minimum-size unavoidable subsets of cyclic permutations

Let $\overline{S_k}$ denote the set of cyclic permutations of length $k$, and let an unavoidable subset be a subset that every sufficiently long permutation contains as a vincular pattern. The sets from the preceding theorem have cardinality

$$(k-1)!.$$

Minimum-cardinality conjecture. For all $k \geq 1$, the minimum cardinality of an unavoidable subset of $\overline{S_k}$ is

$$(k-1)!.$$

The conjecture asserts that the explicitly constructed sets are the most efficient unavoidable subsets. It follows the established construction of unavoidable sets of cardinality $(k-1)!$; the claim that no smaller unavoidable subset exists remains unresolved in the supplied text.