A Leinster group is a finite group $G$, such, that that the sum of orders of its normal subgroups is $2|G|$. A perfect number is a positive integer that is equal to the sum of its proper positive divisors.
Tom Leinster proved that an abelian group $G$ is a Leinster group iff it is cyclic and $|G|$ is a perfect number.
There are two conjectures about those objects:
1) "The number of perfect numbers is infinite." (IPN)
2) "The number of Leinster groups is infinite." (ILG)
Obviously, IPN implies ILG, but does ILG imply IPN, or are those two conjectures not equivalent?