The weak Lang conjecture asserts that rational points on a variety of general type defined over $\mathbb{Q}$ are not Zariski dense (same replacing $\mathbb{Q}$ with a number field). This one is proved in dimension $1$, and in dimension $2$ it would in particular mean that the rational points of surface of general type are contained in the union of a finite number of rational and elliptic curves, plus a finite set of "isolated points".

It seems natural for me that a surface of general type has "less points" because the polynomials that define it are of "higher" degree. However, is there some more precise evidence for this conjecture ?

And in which particular cases is the conjecture known to be true?

I am mostly interested in the case of surfaces, but higher dimension case is also interesting.