Not an answer; just a request for clarification.
[![Tours5][1]][1]
The points are not in convex position. The optimal tour is shown in the left figure.
(1) Is it "$k$-optimal" in your notation for some $k$? "Uniquely $k$-optimal"?
(2) The optimal tour cannot be improved. Do the two suboptimal tours each illustrate "exchanging $2$ edges"? Or $3$ edges? Or $6$?