On 2-adic orders of some binomial sums

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$\sum _{k=-n}^n\left(\genfrac{}{}{0px}{}{2n}{n-k}\right)k^{2r}$$ is divisible by $2^{2n-\min \{\alpha (n),\alpha (r)\}}$, where $\alpha (n)$ denotes the number of 1’s in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.