Weak variant of Mayer's conjecture in dimension one

Let $\Omega \subset \mathbb{C}$ be a bounded domain with

$$\Omega_\mathbb{R}=\Omega\cap\mathbb{R}\neq\emptyset$$

and let $\Phi_k:\Omega\to\Omega$ be contracting holomorphic mappings with unique fixed points $z_k^*\in\Omega_\mathbb{R}$. Suppose that $\Phi_k'(z_k^*)$ is real for every $k$, and define the corresponding transfer operator by

$$(\mathcal{L}f)(z)=\sum_{k=1}^{K}\Phi_k'(z)(f\circ\Phi_k)(z).$$

Weak variant of Mayer's conjecture. All eigenvalues of $\mathcal{L}$ with sufficiently small modulus should be real.

This weakens Mayer's original conjecture that the entire spectrum is real under these hypotheses. The claim is refuted: the analytic maps constructed in the paper provide counterexamples, following counterexamples to Mayer's conjecture given by Levin.