Asymptotic semi-periodicity of interspike intervals for periodic inputs

Let $f\in L^{1}_{\operatorname{loc}}(\mathbb{R})$ and let $\Phi:\mathbb{R}\to\mathbb{R}$ be the firing map of $\dot{x}=f(t)$. For $t\in\mathbb{R}$, define the consecutive-spike differences

$$\eta_n(t)=\Phi^n(t)-\Phi^{n-1}(t).$$

Asymptotic semi-periodicity conjecture. If $f$ is periodic, then for every $t$ the sequence $\eta_n(t)$ is asymptotically semi-periodic. If $f$ is almost periodic, then for every $t$ the sequence $\eta_n(t)$ is asymptotically almost periodic. The conjecture concerns the long-term dynamical behaviour of the interspike intervals generated by the firing map; the source gives no resolution or partial result establishing these assertions.