  Given a finite set $ S $ of points, we consider the following reconfiguration
graph. The vertices are the plane spanning paths of $ S $ and there is an edge
between two vertices if the two corresponding paths differ by two edges (one
removed, one added). Since 2007, this graph is conjectured to be connected but
no proof has been found. In this paper, we prove several results to support the
conjecture. Mainly, we show that if all but one point of $ S $ are in convex
position, then the graph is connected with diameter at most $ 2 | S | $ and
that for $ | S | \geq 3 $ every connected component has at least $ 3 $
vertices.
