Sets in $\mathbb{Z}_m$ whose difference sets avoid squares

We prove that the bound $|A|\leq m^{1/2}(3n)^{1.5n}$ holds for all square-free $m\in\mathbb{N}$ and any set $A\subset\mathbb{Z}_m$ such that $A-A$ contains no nonzero squares, where $n$ denotes the number of odd prime divisors of $m$.
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