Nonagon conjecture for the Cheeger constant at fixed diameter and circumradius

Let $\mathcal{K}^2$ denote the class of planar convex sets, and let $h(\Omega)$, $d(\Omega)$, and $R(\Omega)$ denote respectively the Cheeger constant, diameter, and circumradius. For a constant-width set $K$, let $\Gamma$ and $\gamma$ be its concentric circumcircle and incircle, with

$$d(K)=\omega(K)=R(K)+r(K).$$

Construct the associated nonagon of constant width by inscribing an equilateral triangle $ABC$ in $\Gamma$, taking the circular arcs of radius $R(K)+r(K)$ centered at its vertices and tangent to $\gamma$ at the opposite points, and adding the three circles of radius $(R(K)+r(K))/2$ whose chords are the sides of the triangle and whose centers lie inside it. Nonagon conjecture. For $\Omega\in\mathcal{K}^2$,

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

where $N$ is a nonagon of constant width satisfying $d(N)=d(\Omega)$ and $R(N)=R(\Omega)$. This conjectured extremal property is attributed in the source to the described nonagon construction; the source gives no resolution.