The generalized 2-adic valuation conjecture for weighted Catalan numbers

Let $k$ be a positive integer, and let $L_n^{(k)}$ denote the weighted Catalan numbers with weight $b(x)=(2x+1)^{2k}$. 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.

Generalized 2-adic valuation conjecture. There exists a 2-adic integer $\alpha_k$ and a nonnegative integer $c_k$ such that

$$\xi_2(L_n^{(k)}-C_n)=s_2(n)+\xi_2(n-\alpha_k)+c_k$$

for all $n\geq 2$.

The authors present this as a generalization of Postnikov's conjecture and note that analogous behavior was not observed for other polynomial weight functions; the excerpt gives no proof or resolution.