An Old Conjecture Disproved - and Another Question

Long ago, I had struggled unsuccessfully with this claim:

For any given number N, if any given convex polygonal region P allows a zero-waste partition into N non-convex congruent pieces each with finitely many sides, then P also allows a partition into N convex congruent pieces with zero wastage.

Eg: https://arxiv.org/pdf/1012.3106.pdf

Now, thanks to Prof. Noam Elkies, we have a very neat counter example. See this discussion:

https://mathoverflow.net/questions/370302/a-claim-on-partitioning-a-convex-planar-region-into-congruent-pieces
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A side excursion into cycloids - an area I know very little about - resulted in this discussion: https://mathoverflow.net/questions/370727/on-cycloids-and-other-roulettes