Not an answer; just a request for clarification.
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[![Tours5][1]][1]
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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 $k$?

(2) The optimal tour cannot be improved. Do the two suboptimal tours each illustrate "exchanging $2$ edges"?
Or $3$ edges? Or $6$?


  [1]: https://i.sstatic.net/PDIN4.jpg