Isoceles Triangles - Further Questions

Continuing from this earlier post: https://nandacumar.blogspot.com/2019/10/on-isoceles-triangle-question.html, let me record the following questions:

- Given any 2D convex region C with a mirror symmetry. We need to find the smallest area (likewise smallest perimeter) triangle that contains C. Is it sufficient to only search among isosceles triangles aligned along the direction of mirror symmetry of C for answers to both questions?

- Similarly, if we seek the largest area (largest perimeter) triangle contained within C, is it enough to look only among isosceles triangles aligned along the direction of symmetry of C?

Updates should follow...

Update: Both questions have been answered in the negative by nice and simple counterexamples here: https://mathoverflow.net/questions/345552/smallest-triangles-that-contain-2d-convex-regions-with-reflection-symmetry
Thanks!