On the generation of the symmetric group by index relations of three prefix reversals

The cubic pancake graphs are Cayley graphs over the symmetric group Sn generated by three prefix reversals. There is the folllowing open problem: characterize all the sets of three prefix reversals that generate Sn. Blanco and Golubyatnikov give a complete solution of the problem when any of the two smallest or the two largest lengths but n are included in a triple of prefix reversals. According to their conjectures, we give some conditions implying a triple of prefix reversals does not generate Sn. Moreover, Using the symmetric groups that can already be generated by three prefix reversals, we can derive some recurrence conditions.