Granville gives p.5 an implication of the abc conjecture:

Assume the abc conjecture. Let $f(x,y)$ be squarefree homogeneous polynomial with integer coefficients. For coprime integers $m,n$ if $q^2 \mid f(m,n)$ then $q \ll \max(|m|,|n|)^{2+\epsilon}$.

Can we strengthen this to $q \ll |mn|^{1+\epsilon}$?

For constant $n$ this is consistent with the paper.

Are there heuristic arguments for small roots modulo squares?