For any coprime integers $(x, y)$, $m\geq 2$, $\delta\geq 1$, is it true that $\mid x^m - y^{m+\delta} \mid \geq \delta$?

Question

Is it true that if $x,y,m,\delta$ are integers, $\gcd(x,y)=1$, $m\ge2$, $\delta\ge1$, then $$|x^m-y^{m+\delta}|\ge\delta?$$

Any proofs or references will be most welcome.

Answer 1

$|5^3-2^{3+4}|<4{}{}{}{}{}$.