Is it true that for every $k>0$ Diophantine equation $$ y^2 + x^2y + z^2x + 1 = 0 $$ has an integer solution $(x,y,z)$ such that $\min\{|x|,|y|,|z|\}\geq k$?
Motivation: this equation arises in the study of the smallest open Diophantine equations. If $H$ is the number obtained after replacing all variables by $2$ and all coefficients by the absolute values, then for the given equation we have $H=4+8+8+1=21$. For all equations with $H\leq 20$, I can decide the existence of large solutions, and in fact describe all integer solutions in a certain way. I can also do this for all equations of size $H=21$ with three exceptions: the two equations $2x^2+xyz\pm(y^2+1)=0$ discussed here and this one.
I remark that the equation has infinitely many integer solutions, because, for example, for $x=-145$, the resulting two-variable quadratic equation has infinitely many solutions in integers $(y,z)$. However, this does not answer the question.
