Postnikov's 2-adic valuation conjecture for Morse link and Catalan numbers

Let $L_n$ be the Morse link numbers, let $C_n$ be the Catalan numbers, let $s_2(n)$ denote the sum of the binary digits of $n$, and let $\xi_2$ denote the 2-adic valuation. Let $\alpha=\dotsc 010111_2$ be a 2-adic integer.

Postnikov's 2-adic valuation conjecture. There exists such a 2-adic integer $\alpha$ satisfying

$$\xi_2(L_n-C_n)=s_2(n)+\xi_2(n-\alpha)+2$$

for all $n\geq 2$.

This is attributed to Postnikov and is introduced among conjectures about 2-adic valuations of sequences related to $L_n$; no resolution is supplied in the excerpt.