4-Engel and (2,3,5)-torsion-free implies class seven for groups

Statement
Suppose $$G$$ is a 4-Engel group. Suppose, further, that $$G$$ does not have any 2-torsion, 3-torsion, or 5-torsion, i.e., $$G$$ does not have any non-identity element of order 2, 3, or 5. Then, $$G$$ is a nilpotent group and its nilpotency class is at most seven.

Similar facts for 4-Engel groups

 * 4-Engel implies locally nilpotent

Similar facts for other Engel groups

 * 2-Engel implies class three for groups
 * 2-Engel and 3-torsion free implies class two for groups
 * 3-Engel and (2,5)-torsion-free implies class four for groups