$r$ is parameter. Pick coprime $m,n\in[r,2r]$ with $mn$ even. Consider the Linear Diophantine Equation $$a^4u+b^4v+c^2z=0$$ where $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$.

Is it true that there are constants $$\alpha,\beta,\gamma,\delta>0$$ such that $$|u|,|v|<\alpha r^2\implies|z|>\beta r^6$$ $$|z|<\gamma r^6\implies|u|+|v|>\delta r^2$$ holds?

Related: Small linear relations between primitive Pythagorean triples $\mathsf{II}$

I think above is true for the following reason:

$a^4u+b^4v\bmod c^2$ seems to admit enough room to get $|u|,|v|>c>r^2$. Then since $a^4|u|,b^4|v|>r^10$ then it seems $r^6$ should be the lower bound for $|z$. How to show this formally is unclear to me.