Binary-weight divisibility conjecture for even binomial moments

Let $n,r\geqslant 1$. Define $\alpha(N)$ to be the number of $1$'s in the binary expansion of $N$. Binary-weight divisibility conjecture. The sum

$$\sum_{k=1}^n{2n\choose n-k}k^{2r}$$

is divisible by

$$2^{2n-\min\{\alpha(n),\,\alpha(r)\}-1}.$$

The conjecture refines the evident divisibility by $4^{n-r}$ when $n\geqslant r$, and predicts that the exact power of $2$ also depends on the binary weights of both $n$ and $r$.