The real Collatz co-null liminf conjecture

Let $C$ denote the real Collatz map used in the paper, and let $x\in(3/4,\infty)$. A subset $A$ of a measurable set $B\subseteq\mathbb{R}$ is Lebesgue-co-null in $B$ when $\lambda(B\setminus A)=0$. Real Collatz co-null liminf conjecture. The set

$$\left\{x\in\left(\frac{3}{4},\infty\right)\mathrel{\middle|}\liminf_{n\in\mathbb{N}}C^n(x)=\frac{3}{4}\right\}$$

is Lebesgue-co-null in $(3/4,\infty)$.

This conjecture strengthens the paper’s Lebesgue-measure theorem by proposing the optimal limiting lower bound $3/4$. The corresponding comeager statement is proved, but the Lebesgue-co-null assertion remains open in the supplied text.