Generalized Collatz convergence and cycle conjecture for the triplets $(2^{p+1},3\times2^p,2^p)_+$

For $p\geq 0$, let $(d_p,\alpha_p,\beta_p)_+=(2^{p+1},3\times2^p,2^p)_+$ and define

$$T_p(n)=\begin{cases}n/2^{p+1},&n\equiv0\pmod {2^{p+1}},\\\\(3n+[n]_p)/2,&n\not\equiv0\pmod {2^{p+1}},\end{cases}$$

where $[n]_p$ denotes the remainder in Euclidean division by $d_p=2^{p+1}$. The conjecture has separate assertions for $p\ne2$ and $p=2$.

Generalized Collatz convergence and cycle conjecture. For all $p\geq0$ with $p\ne2$, the triplet $(2^{p+1},3\times2^p,2^p)_+$ is strongly admissible of order one, with unique trivial cycle

$$\Omega(1)=\bigl(1\rightarrow2\rightarrow2^2\rightarrow\cdots\rightarrow2^{p+1}\rightarrow1\bigr),$$

and for every integer $n\geq1$ there exists $k\geq0$ such that $T_p^{(k)}(n)=1$. For $p=2$, $(8,12,4)_+$ is strongly admissible of order two, with trivial cycles $\Omega(1)=(1\rightarrow2\rightarrow4\rightarrow8\rightarrow1)$ and $\Omega(67)=(67\rightarrow102\rightarrow156\rightarrow236\rightarrow356\rightarrow536\rightarrow67)$, and for every $n\geq1$ there exists $k\geq0$ such that $T_2^{(k)}(n)\in\{1,67\}$.

This is a generalized Collatz-type convergence claim: it specifies all asserted trivial cycles and requires every positive integer to reach one of their representatives. The supplied source gives no resolution of these assertions.