The prime-generator index–order conjecture for Collatz processes

Let $f$ be the Collatz function and let $\{f^{s}(b)\}_{s=1}^{\infty}$ be a full Collatz process. The prime-generator index–order conjecture. If the process converges, then

$$b\text{ is prime}\quad\Longleftrightarrow\quad \mathrm{Ind}_f(b)=\tau_f(b)+1.$$

Here $\mathrm{Ind}_f(b)$ is the index and $\tau_f(b)$ is the order of $b$ under the Collatz process. The paper proposes this as a relation between primality and the process invariants; no resolution is supplied.