Periodic oscillation conjecture for combinatorial classes with evenly spaced dominant singularities

Let $\mathcal{C}$ be a combinatorial class with generating function $f_\mathcal{C}(z)$ of radius of convergence $\rho<1$, and suppose

$$f_\mathcal{C}(z)=g(z)+\frac{h(z)}{(1-z/\rho)^\alpha}.$$

Assume $\alpha\notin-\mathbb{N}$, and that $g$ and $h$ are analytic on the cut disk $\overline{D}(0,\rho)\setminus\mathbb{R}^{\leqslant0}$. If the only singularities on the line of convergence of the Dirichlet generating function $F_\mathcal{C}(s)$ are evenly spaced at

$$\log(1/\rho)+2\pi i\mathbb{Z}/\omega$$

for some constant $\omega$, then the periodic oscillation conjecture asserts that

$$|\mathcal{C}_{\leqslant x}|\sim c(x)\rho^{-x}x^{\alpha-1},$$

where $c(x)$ is a bounded function that oscillates with period $\omega$. Classes with irrational object sizes can exhibit such noncontinuous periodic fluctuations, while the conjecture predicts the corresponding asymptotic form under the stated analytic and singularity hypotheses.