Maximum-period conjecture for the logistic map over $\mathbb{Z}_{3^n}$

Let $\mathbb{Z}_{3^n}$ be the ring of residue classes modulo $3^n$, and consider the logistic map

$$f_{3^n}(x)=\mu x(x+1)\bmod 3^n,$$

where $\mu,x\in\mathbb{Z}_{3^n}$. For an initial value $x_0\in\mathbb{Z}_{3^n}$, let $L(x_0;\mu,3^n)$ denote the period of the resulting sequence, and define

$$L(\mu,3^n)=\max\{L(x_0;\mu,3^n)\mid x_0\in\mathbb{Z}_{3^n}\}.$$

Maximum-period conjecture. When $\mu\bmod 3=1$, the maximum period is

$$L(\mu,3^n)=3^{n-2}.$$

The claim concerns the corrected formula for the maximum period of the logistic map over $\mathbb{Z}_{3^n}$; the source states that an earlier representation does not hold in several residue cases and says that this conjecture is proved later in the paper as a corollary.