Yamanouti and nonagon extremal conjecture for the Cheeger constant at fixed minimal width and circumradius

Let $\mathcal{K}^2$ denote the class of planar convex sets, and let $\omega(\Omega)$, $h(\Omega)$, and $R(\Omega)$ denote respectively the minimal width, Cheeger constant, and circumradius. Let $Y$ be a Yamanouti set and $N$ a nonagon of constant width. Yamanouti and nonagon extremal conjecture. For $\Omega\in\mathcal{K}^2$, the following hold:

1. If

$$\omega(\Omega)\in\left[\frac{3}{2}R(\Omega),\sqrt{3}R(\Omega)\right],$$

then

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

where $\omega(Y)=\omega(\Omega)$ and $R(Y)=R(\Omega)$.

2. If $\omega(\Omega)\geq\sqrt{3}R(\Omega)$, then

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

where $\omega(N)=\omega(\Omega)$ and $R(N)=R(\Omega)$. These are further numerical Blaschke--Santaló diagram conjectures; the source supplies no proof or resolution.