Let $S\subseteq\mathbb{T}=\{z\in\mathbb{C}:\vert z\vert=1\}$ with $\text{conv}(S)$ contains $\left\{z\in\mathbb{C}:\vert z\vert\leq\frac{1}{\sqrt{2}}\right\}$. Define $$\mathcal{A}_S:=\{B\in M_2: \Re(\lambda B)\geq-\frac{1}{2}I \text{ for all } \lambda\in S\}.$$ What is $\sup\limits_{S}\sup\limits_{B\in\mathcal{A}_S}\Vert B\Vert$?
Comment: I can see $\sup\limits_{B\in\mathcal{A}_S}\Vert B\Vert<\sqrt{2}$ for fixed $S$ but could not find out the required thing.
Any comment is highly appreciated. Thanks in advance.