The bounded-interval conjecture for the real Collatz map

Let $\operatorname{Col}_R$ denote the real analogue of the Collatz map considered in the paper, and let $r$ be a real number satisfying $r>9/4$. Bounded-interval conjecture. There exists $n>0$ such that

$$\operatorname{Col}_R^n(r)\in\left(\frac{3}{4},\frac{9}{4}\right].$$

The question asks whether every orbit starting above $9/4$ eventually reaches the invariant interval $(3/4,9/4]$. The paper presents this as a more tractable analogue of the original Collatz problem; the stated partial results do not resolve it.