Normalizer covering number 11 conjecture

Let $G$ be a group, and let $\sigma_n(G)$ denote its normalizer covering number, the least number of proper normalizers whose union is $G$, when such a covering exists. Normalizer covering number 11 conjecture. There is no group $G$ with

$$\sigma_n(G)=11.$$

The paper notes that $\sigma_n(S_4)=7$ and asks whether some integer greater than $2$ is not a normalizer covering number; this conjecture proposes $11$ as such an integer.