Clipping Open Paths in Clipper Library

In Clipper, given an open path (black) and a closed polygon (blue), how can I clip the open path such that the clipped parts get completed using the polygon contours and, when possible, a single path remains instead of disjoint segments?
What Clipper operations can achieves this?

Note that the clipping polygon is not necessarily a rectangle like in the example, so the completed segments may be polylines themselves.

Based on this thread it is not possible to do so for open paths.

However, if the open subject path can be closed with a closing path (which is not always guaranteed to be possible), then intersecting/clipping it and then removing the closing path segments will leave the desired results.