Finite primitive permutation groups and regular cycles of their elements

We conjecture that if G is a finite primitive group and if g is an element of G, then either the element g   has a cycle of length equal to its order, or for some r,mr,m and k  , the group G≤Sym(m)wrSym(r)G≤Sym(m)wrSym(r), preserving a product structure of r   direct copies of the natural action of Sym(m)Sym(m) or Alt(m)Alt(m) on k-sets. In this paper we reduce this conjecture to the case that G is an almost simple group with socle a classical group.