Invariance conjecture for points at infinity under changes of cuts

Let $M$ be a complex curve equipped with two families of cuts satisfying the conditions in the preceding conjecture. For each integer $n\geq0$, let the associated set of $C^n$-points at infinity be the boundary data defined from the chosen family of cuts. Invariance conjecture. The two families of cuts lead to the same set of $C^n$-points at infinity for every $n\geq0$. The assertion is intended as a further consequence of the preceding comparison of cut systems; the supplied text gives no resolution of it.