Operator Kantorovich inequality for 2-positive linear maps

Let $0<m\le A\le M$, and let $X,Y$ be partial isometries on a Hilbert space $\mathcal{H}$ whose final spaces are orthogonal to each other. Let $\Phi$ be a $2$-positive linear map. Operator Kantorovich inequality. Then

\left\\|\Phi(X^*AY)\Phi(Y^*AY)^{-1}\Phi(Y^*AX)\Phi(X^*AX)^{-1}\right\\|\le \left(\frac{M-m}{M+m}\right)^2.

This assertion is motivated by the operator Wielandt inequality and related Schwarz inequalities for $2$-positive maps. The authors state that they have been unable to prove or disprove it, so its status remains open.