# Fractional reverse direction Cauchy-Schwarz inequality

If $$Z_1,\dots,Z_r$$ are complex $$m\times m$$-matrices, then let $$\Phi(A_1,\dots,A_r):M_m(\mathbb{C})\rightarrow M_m(\mathbb{C})$$ be the linear mapping defined by $$\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\dots+A_rXA_r^*$$. As always, let $$\rho(Z)$$ denote the spectral radius of an operator $$Z$$.

Suppose that $$A_1,\dots,A_r$$ are $$n\times n$$-complex matrices where $$\Phi(A_1,\dots,A_r)$$ is not nilpotent and $$1\leq d\leq n$$. Then do there always exist $$d\times d$$-complex matrices $$X_1,\dots,X_r$$ where

$$\frac{\rho(A_1\otimes X_1+\dots+A_r\otimes X_r)^2}{\rho(\Phi(A_1,\dots,A_r))\cdot \rho(\Phi(X_1,\dots,X_r))}\geq\frac{d}{n}?$$

My answer here and my other answer show that the answer to this question is yes but only for specific matrices $$A_1,\dots,A_r$$, so I am wondering if this can be generalized. Furthermore, I know that the answer to this question is yes when $$d=n$$ since if $$d=n$$, then one can set $$X_j=\overline{A_j}$$ for $$1\leq j\leq n$$.