Analytic-root conjecture for rational Floquet operators

Let $\alpha=p/q$ be rational with $\kappa>0$, and let $P(t,z)$ be the polynomial constructed from the Floquet operator for the on-resonance double kicked rotor with parameters $\alpha$ and $\kappa$. Its roots are functions of $t$.

Analytic-root conjecture. Each root of $P(t,z)$ is an analytic function of $t$ with period $1$.

If true, this would provide strong support for the paper's broader conjecture that, as rational approximations approach an irrational $\alpha$, the spectrum of the on-resonance double kicked rotor approaches a Cantor set. The source gives numerical motivation but no resolution of this analytic-root claim.