Mersenne-triplet admissibility conjecture

Let $p\geq 2$. The triplet $(2^{p-1},2^{p}-1,1)_{\pmb{+}}$ is an admissible triplet, with order and admissibility as specified below. Its trivial cycle is

$$\Omega(1)=(1\rightarrow 2\rightarrow 2^2\rightarrow\ldots\rightarrow 2^{p-1}\rightarrow 1).$$

Mersenne-triplet admissibility conjecture. For all $p\geq 2$, $(2^{p-1},2^{p}-1,1)_{\pmb{+}}$ is an admissible triplet of order one. It is strongly admissible only for $p=2$ and weakly admissible for all $p\geq 3$.

The case $p=2$ recovers the classical Collatz triplet, while the theorem immediately preceding the conjecture establishes the displayed cycle. The asserted classification of strong versus weak admissibility remains unresolved in the supplied source.