We continue from https://mathoverflow.net/questions/375536/cutting-convex-regions-into-equal-diameter-and-equal-least-width-pieces. There we had asked for algorithms to partition a planar convex polygon into (1) $n$ convex pieces of equal width with the common width maximized and (2) $n$ pieces of equal diameter with the common diameter maximized.  


**Question:** Given any planar convex $C$ and any $n$, if both above questions have somehow been solved (ie we have partitions of $C$ into $n$ convex pieces such that (1) the common width is maximized and (2) common diameter is minimized), will we always have the same partition answering both requirements? I can't find a counter example.