Diameter of reduced spherical convex bodies

The intersection $L$ of two different non-opposite hemispheres of the unit sphere $S^2$ is called a lune. By $\Delta (L)$ we denote the distance of the centers of the semicircles bounding $L$. By the thickness $\Delta (C)$ of a convex body $C \subset S^2$ we mean the minimal value of $\Delta (L)$ over all lunes $L \supset C$. We call a convex body $R\subset S^2$ reduced provided $\Delta (Z) < \Delta (R)$ for every convex body $Z$ being a proper subset of $R$. Our aim is to estimate the diameter of $R$, where $\Delta (R) < \frac{\pi}{2}$, in terms of its thickness.