String C-groups as transitive subgroups of Sym(n)

If $\mathrm{\Gamma }$ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sym(n) (other than Sym(n) and the alternating group Alt(n)), then the rank of $\mathrm{\Gamma }$ is at most $n\mathrm{/}2+1$, with finitely many exceptions (which are classified). It is conjectured that only the symmetric group has to be excluded.