Stable-set conjecture for the real $3x+1$ extension

Let

$$f(x)=x+\frac14-\left(\frac{x}{2}+\frac14\right)\cos(\pi x)$$

be the real extension of the $3x+1$ map on $\mathbb{R}^{+}$, and let $\mu_3=2.445\ldots$ be the repelling fixed point introduced in the paper.

Stable-set conjecture. The function $f$ has no attracting cycle in the interval $[\mu_3,+\infty)$.

Chamberland formulated this conjecture; the paper notes that it implies the no-nontrivial-cycles formulation of the $3x+1$ problem.