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I (OP) take the liberty of updating again this answer, since I realized that defining it "incomplete", as I did, is a bit misleading, letting perhaps one think that there's just some detail to be fixed. In fact, I think that the difficulty described in the Edit above is the key one. I arrived at versions of it through each one of the attempts I made at solving the problem.

I (OP) take the liberty of updating again this answer after several months, since I realized that defining it "incomplete", as I did, is a bit misleading, letting perhaps one think that there's just some detail to be fixed. In fact, I think that the difficulty described in the Edit above is the key one. I arrived at versions of it through each one of the attempts I made at solving the problem.

Edit: the following argument is incomplete since it doesn't prove, as it is, that the direction of $S_x$ remains the same, or at least has vanishingly small perturbations, when $x$ goes to $\infty$. See the comments below.

Edit: the following argument is incomplete insufficient, since it doesn't prove that the direction of $S_x$ remains the same, or at least has vanishingly small perturbations, when $x$ goes to $\infty$. See the comments below.

Update (January 5th, 2023)

I (OP) take the liberty of updating again this answer, since I realized that defining it "incomplete", as I did, is a bit misleading, letting perhaps one think that there's just some detail to be fixed. In fact, I think that the difficulty described in the Edit above is the key one. I arrived at versions of it through each one of the attempts I made at solving the problem.

I believe the implication is true for an arbitrary simple curve $\gamma$ whose both ends escape to infinity. First, let's rescale so that the discs have radius $1$. Denote by $D_r(\vec v)$ the open radius-$r$ disc centered at $\vec v$, where $r=1$ if it's omitted.

Edit: the following argument is incomplete since it doesn't prove, as it is, that the direction of $S_x$ remains the same, or at least has vanishingly small perturbations, when $x$ goes to $\infty$. See the comments below.

I believe the implication is true for an arbitrary simple curve $\gamma$ whose both ends escape to infinity. First, let's rescale so that the discs have radius $1$. Denote by $D_r(\vec v)$ the open radius-$r$ disc centered at $\vec v$, where $r=1$ if it's omitted.