Generalized Siegel conjecture for elliptic curves

Let $C$ be an elliptic curve defined over $\mathbb Q$, embedded in projective space, and let $x_n$ be a fixed coordinate in a coprime integral representative $[x_0,\ldots,x_N]$ of a rational point. For a fixed bound $L$, let $S_n(L)$ be the set of points $Q\in C(\mathbb Q)$ for which $x_n$ is divisible by fewer than $L$ primes. Generalized Siegel conjecture. The set $S_n(L)$ is repelled by $C(\overline{\mathbb Q})$: for every $D\in C(\overline{\mathbb Q})$, on any affine piece of $C$ containing $D$, there is a punctured neighbourhood $N(D)$ of $D$ in the archimedean topology such that

$$N(D)\cap S_n(L)=\emptyset.$$

This generalizes the finiteness phenomenon in Siegel's theorem from coordinates equal to $1$ to coordinates having fewer than a fixed number of prime divisors. The supplied text does not establish the conjecture or provide evidence resolving its status.