The formal power series conjecture equivalent to Collatz

For each $k\geq 1$, let $A_k(x):=\sum_{n\geq 0}a_{k,n}x^n$ be the modified formal power series introduced from the arrival recurrence, and let $A_k'(0)$ denote its derivative at zero. Formal power series conjecture. For all $k\geq 1$, $A_k'(0)=a_{k,1}\neq 0$.

The source states that this condition is equivalent to the Collatz conjecture and gives a formal-power-series reformulation of the arrival dynamics. No resolution status is supplied, so it remains open here.