Spectral-radius refinement of symbolic periodicity

Let $k$ be the period asserted by the symbolic periodicity conjecture. Let ${\bf M}_\pm$ be the matrices associated with the palindromic exponent vectors ${\bf a}_\pm$, with characteristic polynomials $P_\pm$, and let $\rho$ denote spectral radius. For a symbolic period $\epsilon_1,\ldots,\epsilon_k$, define the path-ordered product

$${\mathbf\Pi}={\bf M}_{\epsilon_1}\cdots{\bf M}_{\epsilon_k}.$$

Spectral-radius refinement conjecture. There are two possibilities: (1) if $|\lambda_{\max}|=\rho({\bf M}_\pm)>\rho({\bf M}_\mp)\geq1$, then $k=1$ and the symbol is repeated, $\epsilon_1=\pm$ respectively; or (2) if $|\lambda_{\max}|=\rho({\bf M}_+)=\rho({\bf M}_-)$, then

$$\rho({\mathbf\Pi})=|\lambda_{\max}|^k$$

with $k\geq1$. This is a refinement of the preceding symbolic-periodicity conjecture, relating the eventual period to the dominant spectral radius; it remains open in the supplied source.