Suppose that we have a convex cap, i.e., a surface in $R^3$ homeomorphic to a disk, which lies on the boundary of its convex hull and whose own boundary lies in a plane. Reflect the cap through the plane of its boundary and glue it back to the original cap along the common boundary curve. If the closed surface we obtain is convex, then it is isometrically rigid by a theorem of Pogorelov, i.e., any surface which is isometric to it is congruent to it.

<img src="https://i.sstatic.net/vd4N3.png" width="200">

My question is: what happens when the doubled surface is not convex? Do we still have rigidity?
I do not know the answer even in the case where each cap is a (large) piece of a sphere. Here we want to look at isometric surfaces which are smooth except along the curve where the two pieces meet, so that we do not get trivial examples by reflecting a portion of the surface.

In general very little is known about rigidity of non-convex surfaces, but this situation seems so simple that perhaps someone might have an idea about it, or know of a reference.