The additive-map factorization conjecture for alternative division rings

Let $D$ be an alternative division ring, and let $f,g:D\rightarrow D$ be additive maps. For every $x\in D^\times$, where $D^\times$ denotes the set of invertible elements of $D$, suppose that

$$f(x)+x^2g(x^{-1})=0.$$

Additive-map factorization conjecture. There exists a fixed element $q\in D$ such that

$$f(x)=xq\qquad\text{and}\qquad g(x)=-xq$$

for all $x\in D$. This conjecture concerns the common structure of additive solutions to an inverse functional equation in alternative division rings; the supplied text does not state whether it has been resolved.