In my previous post I asked about the hyperbolicity of the affine surface $S'=\{zw \neq o\}$ in the projective surface $z^2 = P(x) Q(y)$ in $\mathbb{P}^3$, where $P$ and $Q$ are two general polynomials of the same degree $d$. Jason wrote that this affine surface is a finite unbranched cover of the product of two affine hyperbolic curve $(\mathbb{C}\setminus Z(P(x)))\times (\mathbb{C}\setminus Z(Q(y)))$,$(x,y,z) \mapsto (x/z, y/z)$ and therefore $S'$ is also hyperbolic as well (what is the generic condition on polynomials $P$ and $Q$?).//

additionally I could prove that this surface is a surface of general type if $d\ge 5$ and the roots of $P$ and $Q$ are not the same.//

Now assume the equation $z^2 = P(x) Q(y)$ on a number field $K$. I want to ask if the number of $K$ - rational points in the affine part $zw \neq 0$ is finite? I want to deduce this result from Lang conjectures which asserts that:

on a surface of general type the Zariski closure of rational points is a proper sub variety not of general type, i.e. in this case it is a finite union of rational and elliptic curves and a finite singular points.