The refined $3x+1$ conjecture

Let $T_R:\mathbb{Z}\to\mathbb{Z}$ be defined by

$$T_R(n)=\begin{cases}\dfrac{3n}{4},&n\equiv 0\pmod 4,\\[2pt]\dfrac{n-2}{4},&n\equiv 2\pmod 4,\\[2pt]\dfrac{3n+1}{2},&n\equiv 1\pmod 2.\end{cases}$$

For $k\geq 0$, define $T_R^{0}(n)=n$ and $T_R^{k+1}(n)=T_R(T_R^{k}(n))$, and call $(T_R^{k}(n))_{k\geq 0}$ the $T_R$-trajectory of $n$. The refined $3x+1$ conjecture. For every integer $n\geq 0$, the $T_R$-trajectory of $n$ converges to $0$. By the preceding equivalence with the usual $3x+1$ conjecture, this is an alternative formulation of the same unresolved problem, with the transient terms divisible by $3$ and the isolated terms congruent to $1$ modulo $3$ streamlined.