Optimality conjecture for contraction coefficients of non-unital CQ channels

Let $\boldsymbol{\rho}=\frac{1}{2}(I+\boldsymbol{w}\boldsymbol{\cdot}\boldsymbol{\sigma})$ and let $A=\boldsymbol{y}\boldsymbol{\cdot}\boldsymbol{\sigma}$ be the states and observables used to define the contraction coefficients of the non-unital CQ channel $\Phi_{\alpha,\tau}$. For the functions in $\mathcal{K}$ appearing in the bounds

$$\eta_{{\rm \widehat{WY}}}^{\rm Riem}(\Phi) \geq \alpha^2\,\frac{1+\sqrt{1-\tau^2}}{2(1-\tau^2)},$$ $$\eta_{x^{-1/2}}^{\rm Riem}(\Phi) \geq \frac{\alpha^2}{\sqrt{1-\tau^2}},$$ $$\eta_{{\rm BKM}}^{\rm Riem}(\Phi) \geq \frac{\alpha^2}{2\tau}\log\frac{1+\tau}{1-\tau},$$

Optimality conjecture. Equality holds in the three displayed bounds above.

The conjecture would establish the exact contraction coefficients for these metrics and, together with the known monotonicity for CQ channels, imply the corresponding ordering of coefficients under $\kappa_1\leq\kappa_2$. The paper states that a proof for arbitrary choices of $\kappa$ is not easy; no resolution is supplied.