No, this is no proof. Yes, it is probably the way you could make a proof. As long as you can embed the curve in the surface of a smooth ball the answer would be yes by "inflating" the sphere a small amount except for a "trench" which contains the curve. The base of the trench also needs to be modified by being angled to be perpendicular to the plane of a best fit circle to the curve at that point. No time for anything better. But I think that is the outline of how you prove it.