Tightness of the strengthened SDP relaxation in dimension two

Let $\mathcal{F}:= \{x \in \mathbb{R}^2: \|x\| \leq 1,\ \|x\| \leq b^T x\}$ be the two-dimensional feasible space, where $b \in \mathbb{R}^2$ is arbitrary. Let $\mathcal{G}$ denote the convex hull defined in the source, and let $\mathcal{R}_{\mathrm{shor}}$, $\mathcal{R}_{\mathrm{ksoc}}$, and $\mathcal{R}_{\mathrm{cuts}}$ denote the stated relaxations and cuts.

Tightness conjecture. The intersection

$$\mathcal{R}_{\mathrm{shor}} \cap \mathcal{R}_{\mathrm{ksoc}} \cap \mathcal{R}_{\mathrm{cuts}}$$

equals $\mathcal{G}$.

The conjecture is motivated by computational experiments in dimension two, in which the relaxation was exact on nearly all tested instances and the added cuts closed the remaining gaps. Its general validity for arbitrary $b \in \mathbb{R}^2$ is not established in the supplied source.