A group G is called a non-inner nilpotent group, whenever it is nilpotent with respect to a non-inner automorphism. In 2018, all finitely generated abelian non-inner nilpotent groups have been classified. Actually, the authors proved that a finitely generated abelian group G is a non-inner nilpotent group, if G is not isomorphic to cyclic groups Zp1p2...pt and Z, for a positive integer t and distinct primes p1, p2, . . . , pt. In this paper, we make this conjecture that all finite non-abelian p-groups are non-inner nilpotent and we prove this conjecture for finite p-groups of nilpotency class 2 or of co-class 2.

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