A squared Pólya–Szegö and Diaz–Metcalf inequality for positive linear maps

Let $\Phi$ be a unital positive linear map, and let $A$ and $B$ be positive operators satisfying

$$0<m_{1}^{2}\leq A\leq M_{1}^{2},$$

and

$$0<m_{2}^{2}\leq B\leq M_{2}^{2},$$

for positive real numbers $m_{1}\leq M_{1}$ and $m_{2}\leq M_{2}$. Write $A\sharp B$ for the operator geometric mean. Squared Pólya–Szegö and Diaz–Metcalf inequalities. The inequalities

$$(\Phi(A)\sharp\Phi(B))^2\leq\frac{1}{4}\left(\sqrt{\frac{M_1M_2}{m_1m_2}}+\sqrt{\frac{m_1m_2}{M_1M_2}}\right)^2\Phi(A\sharp B)^2$$

and

$$\left(\frac{M_2m_2}{M_1m_1}\Phi(A)+\Phi(B)\right)^2\leq\left(\frac{M_2}{m_1}+\frac{m_2}{M_1}\right)^2\Phi(A\sharp B)^2$$

should hold. These inequalities refine operator versions of the Pólya–Szegö and Diaz–Metcalf inequalities by relating the geometric mean of the images, and a weighted sum of the images, to the image of the geometric mean. The supplied text does not give resolution evidence beyond presenting the result as a conjecture, so its status is left open.