I am a complex geometer trying to parse Vojta's conjecture on rational points, and I have a very basic misunderstanding (I apologize if this is too easy for MO).

Let $X$ be a variety over a number field. The conjecture states that for all $P$ outside of some Zariski closed subset,

$\sum_{v \in S} \lambda_{D,v}(P) + h_{K_X}(P) \leq \epsilon h_H(P) + C$,

where $D$ is an effective divisor, the $\lambda_{D,v}$ are a finite set of local heights, the $h$'s are Weil heights, $H$ is an ample class, and $\epsilon$ and $C$ are constants.

Here is my confusion. Let's say $X$ is a K3 surface, and take $D = H$. The conjecture is $\sum_{v \in S} \lambda_{H,v}(P) \leq \epsilon h_H(P) + C$, which means that $\frac{\sum_{v \in S} \lambda_{H,v}(P)}{h_H(P)} \leq \epsilon + \frac{C}{h_H(P)}$.

Suppose that $C \subset X$ is a rational curve, also defined over our number field. We can pull back the above inequality to $\mathbb P^1$ by functoriality of the various height functions. Now, I don't see why the left side is supposed to be small at all for every $P$, never mind less than epsilon. Aren't there points on $\mathbb P^1$ for which most of the height is accounted for by local heights in $S$? (For example, say $S$ contains only the $2$-adic absolute value, and consider points whose coordinates are powers of $2$.) I would expect that this inequality fails for a dense set of points on $C$ (for any $\epsilon < 1/2$, say).

This itself isn't obviously a problem: $C$ would just need to be in Zariski closed set that we exclude. But $X$ might have infinitely many rational curves defined over the number field, and the argument applies to all of them. And this would contradict the conjecture. So I suspect I am misunderstanding what local heights are on $\mathbb P^1$. Can someone straighten me out?