The distinct-count conjecture for supercharacter theories of groups with a fixed number of conjugacy classes

Let $\Gamma(k)$ denote the class of groups considered in the paper having $k$ conjugacy classes, and let $|Sup(G)|$ denote the number of supercharacter theories of a finite group $G$.

Distinct-count conjecture for $\Gamma(k)$. The numbers $|Sup(G)|$ are distinct for all members $G$ of $\Gamma(k)$ if and only if $k=6$. When $k=6$, the groups are

$$Z_5,\ D_{14},\ A_5,\ Z_5:Z_4,\ Z_7:Z_3,\ S_4,\ D_8,\text{ and }Q_8.$$

The conjecture is suggested by the computations summarized in Table 6. The supplied text does not provide a proof or disproof.