3 problems

Interval-rank conjecture. The group $A_{11}$ is the only finite simple group whose set of ranks of string C-group representations is not an interval in the set of integers.

Constant-count conjecture. The number of string C-group representations of rank $n-i$ for $S_n$ with $1\leq i\leq (n-3)/2$ is a constant independent of $n$.

Let $n\geq 12$. An abstract regular polytope with automorphism group $\mathrm{A}_n$ has rank $d$, where rank means the number of involutory generators in its string C-group represe…