Axiality of planar convex regions - alternative definitions

We continue the the previous post:

Here is how axiality of a planar convex region is defined in Wiki:
Axiality is a measure of how much axial symmetry a shape has. It is defined as the ratio of areas of the largest axially symmetric subset of the shape to the whole shape. Equivalently it is the largest fraction of the area of the shape that can be covered by a mirror reflection of the shape (with any orientation).
A shape that is itself axially symmetric, such as an isosceles triangle, will have an axiality of exactly one, whereas an asymmetric shape, such as a scalene triangle, will have axiality less than one. Lassak (2002) showed that every convex set has axiality at least 2/3. This result improved a previous lower bound of 5/8 by Krakowski (1963). The best upper bound known is given by a particular convex quadrilateral, found through a computer search, whose axiality is less than 0.816.

Alternative definitions for axiality:
1. In the above relation, replace "largest axisymmetric subset" by "smallest axisymmetric superset". No idea what can then be said about the shape with least axiality with this new definition. Replacing "area" by "perimeter" could also be considered.
2. In the spirit of Yaglom and Boltyanski, one can define an 'axiality ratio' for every chord L of a planar convex region C as the minimum of the ratio between length of the smaller to the larger segment into which L cuts any chord of C that is perpendicular to itself. Now, the best axis is that chord that maximizes this axiality ratio and that value of the ratio could be called the axiality of C itself.
Question: What shapes minimize axiality under this new definition(s)?

Update (Jan 17th, 2023). Question 2 above has a very simple answer as given here: https://mathoverflow.net/questions/438451/on-axiality-of-planar-convex-regions. Thanks to Prof. Anton Petrunin