Triviality conjecture for jointly $T_a$- and analytic $f$-invariant sets

Let $T_a$ be the fixed transformation, and let $f$ be a real-analytic function on either the real line or the circle:

$$f\in C^{\omega}(\mathbb{R})\quad\text{or}\quad f\in C^{\omega}(\mathbb{R}/\mathbb{Z}).$$

Assume that $f$ is not affine. A set is jointly $T_a$- and $f$-invariant when it is invariant under both transformations. Triviality conjecture. Every jointly $T_a$- and $f$-invariant set is trivial.

The conjecture concerns the rigidity of sets invariant under both an expanding map and a non-affine analytic function. The surrounding results show that, for piecewise curved functions, jointly invariant sets are either the whole interval or have dimension zero, but the stated analytic assertion remains open here.