Generic unbounded distortion for non-Dini moduli of continuity

Let $K$ be the class of moduli of continuity, and let $\Gamma_{\omega}^{1}(\mathbb S^1)$ denote the corresponding class of $C^1$ expanding circle maps with modulus $\omega$. A modulus $\omega$ is non-Dini-integrable if its Dini integral diverges. A map has unbounded distortion if its distortion constants over injectivity domains of all iterates are not uniformly bounded.

Generic unbounded-distortion conjecture. For every non-Dini-integrable $\omega\in K$, unbounded distortion is $C^1$-generic in $\Gamma_{\omega}^{1}(\mathbb S^1)$.

The preceding discussion explains that Dini-integrability guarantees bounded distortion, whereas the paper's construction of absolutely continuous invariant probability measures in the non-Dini setting does not use distortion estimates. This conjecture asks whether, despite the existence of such measures, unbounded distortion is the generic behaviour for every non-Dini-integrable modulus.