Sufficiency of the bow-tie parameter region for maximum step-size coefficient

Consider the generic ERK(3,3) methods in the family defined by, with parameters $(\alpha,\beta)\in\mathbb{R}^2$ satisfying the relevant restriction. Define

$${\cal{B}}:=\left\{ (\alpha,\beta)\in\mathbb{R}^2: \left(\frac{1}{2}\leq \alpha <\frac{2}{3}\land \frac{2}{3}\leq \beta \leq 1-\frac{\alpha}{2}\right)\lor \left(\frac{2}{3}<\alpha \leq 1\land 1-\frac{\alpha}{2}\leq \beta \leq \frac{2}{3}\right)\right\}.$$

Maximum step-size conjecture. All methods in this family with $(\alpha,\beta)\in{\cal{B}}$ have maximum step-size coefficient $\gamma=1$. This claim is intended to characterize the methods in the generic ERK(3,3) family attaining the maximum coefficient. The preceding calculations establish that ${\cal{B}}$ is necessary; the sufficiency assertion is the remaining part of the proposed characterization, and its resolution is not supplied here.