Polynomial Bézout identity for consecutive q-powers

Let $n$ and $r$ be positive integers. Polynomial Bézout conjecture. There exist polynomials $A_{n,r}$ and $B_{n,r}$ in $q$ with integer coefficients such that

$$A_{n,r}(1-q^n)^r+B_{n,r}(1-q^{n+1})^r=(1-q)^r.$$

This conjecture concerns an integral Bézout identity related to the preceding polynomial-coprimality argument; no resolution is given in the supplied text.