Optimality conjecture for the family of positive maps $\Phi[\alpha]$

For the family of positive maps $\Phi[\alpha]$ considered in the paper, with parameter

$$\frac{\pi}{3}\leq\alpha\leq\frac{5}{3\pi},$$

consider the corresponding maps in the boundary segment studied in the structural physical approximation analysis.

Optimality conjecture for $\Phi[\alpha]$. For $\frac{\pi}{3}\leq\alpha\leq\frac{5}{3\pi}$, the positive maps $\Phi[\alpha]$ are optimal.

The claim extends the known optimality of the Choi and reduction maps to the remaining points of this boundary family. The paper indicates that the relevant spanning-vector argument is more complicated away from the reduction-map point, and the supplied text does not establish a resolution.