The binomial alternating-sum gcd conjecture

For positive integers $m$ and $n$, define the sequence of integers

$$A_r(n)=\sum_{k=-n}^n(-1)^k{2n\choose n+k}^r.$$

The alternating-sum gcd conjecture. For all positive $m$ and $n$,

$$\gcd\bigl(A_m(n),A_{m+1}(n),\ldots\bigr)={2n\choose n}.$$

Here the greatest common divisor is taken over all exponents $r=m,m+1,\ldots$. This is presented as one of the paper’s open problems, and no general proof or disproof is supplied.