Logarithmic growth conjecture for integral points on the cubic surface $S_0$

Let $S_0$ be the cubic surface introduced earlier in the paper, and consider its integral points with coordinates satisfying $1\leq x,y\leq N$. A growth conjecture asserts that the number of such integral points grows like

$$c\log^2 N.$$

This concerns the asymptotic distribution of integral points on $S_0$; the source proposes the logarithmic-squared growth rate, but gives no resolution of the conjecture.