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 $1\leq j\leq n$.
