Guo–Jouhet–Zeng gcd conjecture for alternating binomial sums

For positive integers $m$ and $n$, define

$$A_r(n)=\sum_{k=0}^{2n}(-1)^k\binom{2n}{k}^r.$$

Here $\gcd(a_1,a_2,\ldots)$ denotes the greatest common divisor of the indicated integers. Guo–Jouhet–Zeng's gcd conjecture.

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

Calkin's congruence shows that $\binom{2n}{n}$ divides every term in the gcd. The conjecture asserts that this common divisor is exact for every positive $m$ and $n$; the supplied source does not indicate a resolution.