If $A$ is a matrix, then let $\rho(A)$ denote the spectral radius of $A$. If $A=(a_{i,j})_{i,j}$, then let $\overline{A}=(\overline{a_{i,j}})_{i,j}$.

Suppose that $A_1,\dots,A_r\in M_{n}(\mathbb{C})$ are complex matrices where $A_1,\dots,A_r$ has no non-trivial invariant subspace. Let $(X_1,\dots,X_r)\in M_{d}(\mathbb{C})^r$ be complex matrices such that the quantity $$\frac{\rho(A_1\otimes X_1+\dots+A_r\otimes X_r)}{\rho(X_1\otimes\overline{X_1}+\dots+X_r\otimes\overline{X_r})^{1/2}}$$ is maximized. If $d\leq n$, then is $\text{Rank}(X_j)\leq\text{Rank}(A_j)$ for $1\leq j\leq r$?

My computer calculations suggest that the answer to this question is *yes*. This answer and this answer give special cases where the answer to this question is *yes*.

If we drop the assumption that $(A_1,\dots,A_r)$ has no invariant subspace, then the answer to this question is *no* (as a trivial example, let $A_1=\dots=A_r=0_d$ and let $X_1,\dots,X_r$ be arbitrary matrices in $M_d(\mathbb{C})^r$; one can come up with less trivial examples as well).