Yamanouti-set conjecture for the Cheeger constant, perimeter, and minimal width

Let $\mathcal{K}^2$ denote the class of planar convex sets, and let $\omega(\Omega)$, $h(\Omega)$, and $P(\Omega)$ denote respectively the minimal width, Cheeger constant, and perimeter of $\Omega$. A Yamanouti set is the extremal set described in the source. Consider the diagram $(\omega,h,P)$. Yamanouti-set conjecture. If

$$\pi\omega(\Omega)\leq P(\Omega)\leq 2\sqrt{3}\,\omega(\Omega),$$

then

$$h(\Omega)\leq h(Y),$$

where $Y$ is a Yamanouti set such that $P(Y)=P(\Omega)$ and $\omega(Y)=\omega(\Omega)$. This is one of the conjectures suggested by numerical approximations of Blaschke--Santaló diagrams; its resolution is not supplied in the source.