Smallest n-ellipses that contain convex polygonal regions

This post adds some speculations to https://mathoverflow.net/questions/403370/smallest-3-ellipses-that-contain-triangles.

Are the following claims valid?

"Given any convex polygonal region C, and integer n > m, the smallest n-ellipse that contains C is strictly smaller than the smallest m-ellipse that contains C. And the largest n-ellipse contained entirely in C is strictly larger than the largest m-ellipse contained within C."

Here "larger(smaller)" means larger(smaller) both in terms of area and perimeter.

Remarks: The claims above have exceptions. Indeed, if C is a square, the inscribed circle is both the largest 1-ellipse and largest 2-ellipse contained within C. However, it might well be the case that for larger n, larger n-ellipses contained within the square exist - and if such is not the case, an explanation would be needed. Guess: n-ellipses that are tangent to C at the midpoint of each of C's edges seem good candidates for large inscribed objects.

Refs: https://en.wikipedia.org/wiki/N-ellipse https://web.archive.org/web/20160928200222/http://renyi.mta.hu/~p_erdos/1982-18.pdf