Postnikov's periodicity conjecture for weighted Catalan numbers modulo powers of 3

Let $L_n$ denote the Morse link numbers. For a positive integer $r$, consider the sequence of their residues modulo $3^r$.

Postnikov's periodicity conjecture. For $r\geq 3$, the sequence $\{L_n\pmod{3^r}\}$ is purely periodic with period $2\cdot 3^{r-3}$.

The paper presents this as a partial resolution of a conjecture of Postnikov; the stated periodicity is not established in the supplied context.