Is the Bonnesen-Fenchel Conjecture true?

[Guest]

FYI:

"Bonnesen and Fenchel conjectured that Meissner tetrahedra are the minimum-volume three-dimensional shapes of constant width, a conjecture which is still open. In connection with this problem, Campi, Colesanti and Gronchi showed that the minimum volume surface of revolution with constant width is the surface of revolution of a Reuleaux triangle through one of its symmetry axes."

Source Links: https://en.wikipedia.org/wiki/Surface_of_constant_width#CITEREFBonnesenFenchel1934;

https://en.wikipedia.org/wiki/File:ReuleauxTetrahedron_Animation.gif.