Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$

Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$

If $a$ and $b$ are two coprime integers then prove that $\gcd((a+b)^m,
(a-b)^m) \leq 2^m$
My attempt:
I took two cases first one in which a and b are both odd then very clearly
the gcd will be 2^m but I'm stuck in the second case in which one of the
integer is even and the other one is odd I am reaching no where I tried
everything even binomial expansion but I'm stuck.