Conjecture that the automorphism group of a non-cyclic group of prime power order is always bigger

Statement
Let $$p$$ be a prime and $$G$$ be a non-cyclic $$p$$-group; in other words, $$G$$ is a group of prime power order. The conjecture states that the order of the automorphism group $$\operatorname{Aut}(G)$$ is at least equal to the order of $$G$$.