Generic maximal-growth trajectory conjecture for switching systems

Let a switching system have joint spectral radius $\rho=1$, and let $L$ be its resonance degree. For a discrete-time trajectory $x(k)$, write $\\|x(k)\\|$ for its norm; in continuous time use $x(t)$ and the corresponding time variable.

Maximal-growth trajectory conjecture. For a generic discrete-time system, there exists a trajectory $x(k)$ such that

$$\limsup_{k\to\infty}\frac{\log\|x(k)\|}{\log k}=L.$$

The same assertion holds for generic continuous-time systems, with the discrete-time variable replaced by continuous time. It complements the conjectured two-sided growth estimate by asserting that a single trajectory realizes the resonance-degree exponent.