Postnikov's 5-adic valuation conjecture for Morse link numbers

Let $L_n$ denote the Morse link numbers, and let $\xi_5$ denote the 5-adic valuation. Let $\alpha=\dotsc 111111120_5$ be a 5-adic integer.

Postnikov's 5-adic valuation conjecture. For $n\geq 4$,

$$\xi_5(L_n)= \begin{cases} 2 & \text{if } n \text{ is even},\\ \xi_5(n-\alpha)+3 & \text{if } n \text{ is odd}. \end{cases}$$

This is attributed to Postnikov; the source notes that the original formulation contains typographical errors, but supplies no resolution of the corrected statement.