Literature review of a (possibly) open differential geometry problem

Question

I recently came across an apparently simple-sounding problem in basic differential geometry, as mentioned below,

Problem

Let $S \subset \mathbb{R}^3$ be a closed surface of diameter $d$. Suppose that there exists a constant $h < d$ so that whenever a pair of planes separated by a distance of $h$ intersects $S$, the area of $S$ contained between these planes is constant. Does it then follow that $S$ is a sphere?

I searched for references/reviews regarding progress made for this problem, but could not find anything relevant. It would be helpful if someone could point out any relevant material/concepts regarding this problem.

Answer 1

If you restrict to surfaces of revolution, you should look at Andrew Hwang's paper "A Symplectic Look at Surfaces of Revolution," L'Enseignement Mathématique, 49 (2003), 157-172.