Coordinate equivalence conjecture for transformed polynomials

Let $A$ be a commutative ring with one, let $A^{[n]}$ be a polynomial ring over $A$, let $a,b\in A^{[n]}$, and let $g\in A^{[n]}[y]$. Coordinate equivalence conjecture. If $A$ is a $\mathbb{Q}$-algebra, then

$$y+g(ay+b)\text{ is a coordinate}$$

if and only if

$$y+ag(y)-b\text{ is a coordinate}.$$

This question is motivated by the preceding corollary, which establishes the analogous equivalence for being a hyperplane. The supplied text gives no resolution of the coordinate statement.