The Möbius-nonvanishing conjecture for Collatz trajectories of primes

Let $f$ be the Collatz function, let $p>2$ be prime with $p\equiv 3\pmod 4$, and let $\mu$ denote the Möbius function. The Möbius-nonvanishing conjecture. There exists an integer $n$ in the Collatz trajectory of $p$ such that

$$n\in\{f^{s}(p)\}_{s=1}^{\infty}\quad\text{and}\quad\mu(n)\neq 0.$$

This is one of the paper's proposed primality-related properties of Collatz processes. The supplied text gives no resolution evidence.