Monotonicity of asymptotic outbreak size for short-period infection rates

Let $A\in [0,1)$, $\omega>0$ and $\delta\in[0,1]$. Define

$$\beta_t=1+A\sin((t+\delta \omega) 2\pi/\omega).$$

Short-period monotonicity conjecture. There exists $\bar{\omega}>0$ such that, for all $\omega<\bar{\omega}$, the asymptotic outbreak size $1-s^{(\infty)}(\infty)$ is decreasing in $A$.

This conjecture concerns periodic infection rates with fixed recovery rate $\rho\equiv 1$. It formalizes the observed numerical decrease of outbreak size with infection-rate amplitude when the period is sufficiently short; the statement is left for future investigation.