9 problems

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

Alternating-group supercharacter theory conjecture. For all $n\geqslant 7$, every supercharacter theory of $A_n$ other than $\mathsf{M}(A_n)$ can be constructed from a group of tab…

Automorphism refinement conjecture. Then

Three-subgroup closure conjecture. For every $g\in G\setminus\bigcup_iH_i$, the set $[g]_{r^2}$ lies in $\mathscr{K}^{\infty}(\mathcal{C})$.

Automorphism conjecture. If $\mathsf{S}_H\neq\mathsf{M}(H)$, then every subgroup of $G$ is $\mathsf{S}$-normal. In particular, $\mathsf{S}$ comes from automorphisms.

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

Uniqueness and ATLAS-ordering conjecture. For each divisor $r$ of $p-1$, there exists exactly one supercharacter theory $(\mathcal{X},\mathcal{K})$ of $Z_p$ such that every non-tri…

Bad-part density limit conjecture for groups of order $3q_n$.

Bad-part density limit conjecture.