The symmetric-group supercharacter theory conjecture

Let $S_n$ be the symmetric group on $n$ letters. A supercharacter theory of $S_n$ is a supercharacter theory in the sense used for finite groups.

Symmetric-group supercharacter theory conjecture. For every $n\geqslant 7$, the symmetric group $S_n$ has exactly $4$ supercharacter theories.

The conjecture is closely related to the proposed classification for alternating groups and is suggested by the computations described in the source. Its assertion for all $n\geqslant 7$ remains open.