Frobenius conjecture on nth roots

Statement
Suppose $$G$$ is a finite group and $$n$$ is a natural number dividing the order of $$G$$. Suppose the number of $$n^{th}$$ roots in $$G$$, i.e., the number of elements $$g \in G$$ such that $$g^n = e$$, is exactly $$n$$.

The Frobenius conjecture on nth roots states that in that case, that set of $$n^{th}$$ roots must be a subgroup of $$G$$.

Related facts

 * Exactly n elements of order dividing n in a finite solvable group implies the elements form a subgroup: This is precisely the Frobenius conjecture in the case of finite solvable groups.
 * Number of nth roots is a multiple of n
 * Number of nth roots of any conjugacy class is a multiple of n
 * At most n elements of order dividing n implies every finite subgroup is cyclic