The diagonal generalized Collatz conjecture

For $p\geq0$, let $(d_p,\alpha_p,\beta_p)_+=(2^{p+1},3\cdot2^p,2^p)_+$ and let $T_p$ be its associated mapping. Diagonal generalized Collatz conjecture. If $p\geq0$ and $p\neq2$, the triplet is strongly admissible of order one, with unique cycle

$$\Omega(1)=(1\to2\to2^2\to\cdots\to2^{p+1}\to1),$$

of length $p+2$, and every $n\geq1$ satisfies $T_p^{(k)}(n)=1$ for some $k\geq0$. For $p=2$, the triplet $(8,12,4)_+$ is strongly admissible of order two, with cycles $\Omega(1)=(1\to2\to4\to8\to1)$ and $\Omega(67)=(67\to102\to156\to236\to356\to536\to67)$, and every $n\geq1$ satisfies $T_2^{(k)}(n)\in\{1,67\}$ for some $k\geq0$. This is a special case of the generalized conjecture above and includes the classical Collatz mapping when $p=0$; the paper offers computational support but no proof.