The 3-adic valuation pattern conjecture for Morse link numbers

Let $L_n$ denote the Morse link numbers, and let $\xi_3$ denote the 3-adic valuation. For odd $n\geq 3$, if $\alpha$ is a 3-adic integer, define the final ternary digit of the unit part of $n-\alpha$ by

$$\frac{n-\alpha}{3^{\xi_3(n-\alpha)}}.$$

This digit is $1$ or $2$.

3-adic valuation pattern conjecture. There exists a 3-adic integer $\alpha$ such that, for odd $n\geq 3$, the quantity $\xi_3(L_n-1)$ depends only on $\xi_3(n-\alpha)$ and the last digit of

$$\frac{n-\alpha}{3^{\xi_3(n-\alpha)}}.$$

The claim is motivated by computational evidence, and the source explicitly says that the pattern may be more complicated than the analogous 5-adic conjecture; no resolution is supplied.