The variable-exponent central power-difference conjecture for rings

Let $R$ be a ring. For every $x\in R$, choose positive integers $a=a(x)$ and $b=b(x)$, depending on $x$, such that $x^a-x^b$ is central, and such that either $b=1$ or $\gcd(a,b)=1$ with $a$ and $b$ of opposite parity.

Variable-exponent central power-difference conjecture. Under these conditions, $R$ is commutative.

This conjecture extends fixed-degree commutativity results for rings satisfying central power-difference identities. The preceding fixed-exponent corollary establishes the result when the exponents are fixed; allowing $a$ and $b$ to depend on $x$ is the remaining issue.