If $X$ is a complex matrix, then let $\overline{X}=(X^T)^*=(X^*)^T$ (this means that $\overline{X}$ is the matrix obtained by replacing every entry in $X$ with its complex conjugate), and let $\rho(X)$ denote the spectral radius of $X$. Let $M_d(\mathbb{C})^{r++}$ be the set of all tuples $(X_1,\dots,X_r)\in M_d(\mathbb{C})^r$ with $\rho(X_1\otimes\overline{X_1}+\dots+X_r\otimes\overline{X_r})>0.$
If $A_1,\dots,A_r$ are $n\times n$-complex matrices, and $d$ is a natural number, then define $$\rho_{2,d}(A_1,\dots,A_r)=\max\{\frac{\rho(A_1\otimes X_1+\dots+A_r\otimes X_r)}{\rho(X_1\otimes\overline{X_1}+\dots+X_r\otimes\overline{X_r})^{1/2}}\mid(X_1,\dots,X_r)\in M_d(\mathbb{C})^{r++}\}.$$
If $A_1,\dots,A_r$ are $n\times n$-complex matrices where $(A_1,\dots,A_r)$ has no non-trivial irreducible subspace and $1\leq d<g\leq n$, then do we necessarily have $\rho_{2,d}(A_1,\dots,A_r)<\rho_{2,g}(A_1,\dots,A_r)$?
By this answer, we know that $\rho_{2,n-1}(A_1,\dots,A_r)<\rho_{2,n}(A_1,\dots,A_r)$ if and only if $(A_1,\dots,A_r)$ has no non-trivial irreducible subspace.
This is a follow up of this simpler question where for all $n$, I gave a specific example of matrices $X_1,\dots,X_r\in M_n(\mathbb{C})^r$ with $\rho_{2,1}(X_1,\dots,X_r)<\dots<\rho_{2,n}(X_1,\dots,X_r)$.
