the number of different least positive residues modu $p^2$ of $x^{p-1}$ is $= p+1$?

Question

Conjectures:

Let $n$ be postive integers, Prove or disprove there exist infinitely many prime numbers $p$,such the number of different least positive residues modu $p^2$ of $x^{p-1}$ is $= p+1$?

I don't know if the question is correct. Feel this problem using Fermat's little theorem.

I know if $(x,p)=1$,we have $x^{p-1}\equiv 1\pmod p$,if $(x,p)>1$,then $x^{p-1}\pmod p$ the residues have the same value.