Optimality conjecture for the equidistant vector set in even dimensions

Let $m\geq 4$ be an even integer, let $\boldsymbol{E}_m$ denote the vector set defined in the paper, and let $\beta_{\boldsymbol{A}}$ be the condition number associated with $\boldsymbol{A}\in\mathbb{R}^{m\times 2}$. Optimality conjecture.

$$\boldsymbol{E}_m\in\operatorname*{\arg\,\min}_{\boldsymbol{A}\in\mathbb{R}^{m\times 2}}\beta_{\boldsymbol{A}}.$$

The preceding results establish this optimality for odd $m\geq 3$ by matching a general lower bound with the value attained by $\boldsymbol{E}_m$. The conjecture proposes that the same equidistant vector set is optimal when $m$ is even, a case not settled by those results.