Optimality conjecture for harmonic frames in real two-dimensional phase retrieval

Let $m\geq 4$ be an even integer. For $\boldsymbol{A}\in\mathbb{R}^{m\times 2}$, let $\beta_{\boldsymbol{A}}^{\mathbb{R}}$ denote its condition number, and let $\boldsymbol{E}_m\in\mathbb{R}^{m\times 2}$ be the harmonic frame

$$\boldsymbol{E}_m:=\left(\begin{array}{cccc} 1 & \cos \frac{1}{m}\pi & \cdots & \cos \frac{m-1}{m}\pi \\ 0 & \sin \frac{1}{m}\pi & \cdots & \sin \frac{m-1}{m}\pi \end{array}\right)^{T}.$$

Harmonic-frame optimality conjecture. If $m\geq 4$ is even, then

$$\beta_{\boldsymbol{E}_m}^{\mathbb{R}}=\min_{\boldsymbol{A}\in\mathbb{R}^{m\times 2}}\beta_{\boldsymbol{A}}^{\mathbb{R}}.$$

The corresponding equality is known for odd integers $m\geq 3$, while the conjecture asserts optimality of the harmonic frame in the remaining even case.