The Sophie Germain prime distribution conjecture for backward Collatz translates

Let $f$ be a Collatz function and let $\{f^{s}(b)\}_{s=1}^{\infty}$ be the corresponding Collatz process. A process is full in the sense used by the source, and define

$$\mathcal{M}=\{\mathrm{Inf}\{f^{-s}(b)\}-1\}_{s=1}^{\infty}.$$

Let $\rho$ be the set of all primes. The backward-Collatz prime-density conjecture. If the process is full, then $\mathcal{M}$ contains infinitely consecutive primes and

$$\lim_{s\longrightarrow\infty}\frac{\#(\mathcal{M}\cap\rho)}{\#\mathcal{M}}=1.$$

The paper explicitly relates this conjecture to the unresolved distribution problem for Sophie Germain primes.