Periodic switching stability implies absolute stability for pairs in dimension at least four

Let $\boldsymbol{S}=\{S_1,S_2\}\subset\mathbb{R}^{d\times d}$, with $d\geq 4$, be an arbitrary pair satisfying condition (1.3a), namely that the matrices share a common non-strict Lyapunov matrix. A system is periodically switched stable if $\rho(S_{w_n}\dotsm S_{w_1})<1$ for every finite word $(w_1,\dotsc,w_n)\in\{1,2\}^n$ and every $n\geq 1$, and it is absolutely stable if $\rho(\boldsymbol{S})<1$. Periodic switching stability conjecture. If $\boldsymbol{S}$ is periodically switched stable, then it is absolutely stable. Equivalently, if $\rho(\boldsymbol{S})=1$, there exists at least one word $(w_1,\dotsc,w_n)\in\{1,2\}^n$ for some $n\geq 1$ such that

$$\sqrt[n]{\rho(S_{w_n}\dotsm S_{w_1})}=1.$$

This problem concerns the relationship between periodic and arbitrary switching for pairs of discrete-time linear systems with a common non-strict Lyapunov matrix. The paper establishes related stability and spectral finiteness results in lower dimensions, while the asserted implication for $d\geq 4$ is posed for further study.