The generalized Collatz conjecture for the $m=b-1$ family

Let $b$ and $m$ be the parameters defining the generalized Collatz map, and let $(S_k)$ be the sequence obtained by iterating this map from an arbitrary positive integer. Generalized Collatz conjecture. If $m=b-1$, the sequence $(S_k)$ shall eventually reach the number $1$, irrespective of which positive integer is chosen initially.

This conjecture extends the original Collatz assertion to the specified parameter family. The source reports computational searches and counterexamples for several other parameter choices, but gives no proof for the case $m=b-1$; its status is therefore open.