A Variant of the Borsuk Problem - Least Width

Definitions: The diameter of a convex region is the greatest distance between any pair of points in the region. The least width of a 2D convex region can be defined as the least distance between any pair of parallel lines (in higher dimensions, hyperplanes) that touch the region.

The following question is a variant of the Borsuk problem which focusses on diameter (https://en.wikipedia.org/wiki/Borsuk%27s_conjecture):

Question: Can every planar convex region with least width = 1 be partitioned into 4 convex pieces such that every piece has least width at least 1/2?

Remarks: If the answer to above is "yes", one can ask the generalization: can every d-dimensional convex region with least width = 1 be cut into 2^d convex pieces such that each piece has least width at lest 1/2? If the answer is "no", one can ask for a better bound on least width.

One can also ask, given a general fraction f, to find that number of pieces such that the least width among them is at least f when any convex body with unit least width is partitioned; and perhaps try to extend the Borsuk problem itself by asking given an f, at least how many pieces are needed so that the diameter of each piece is less than f.

In 2D, we can also look at quantities such as moment of interia about an axis perpendicular to the plane thru center of mass and see how this it can have a bound among convex pieces.

Mathoverflow page: https://mathoverflow.net/questions/384904/a-variant-of-borsuk-problem-based-on-least-width