The split-octonion extension of Theorem 5.1

Let $O$ be a split-octonion algebra, and let $f:O\rightarrow O$ be an additive map. Write $O^\times$ for the set of invertible elements of $O$. Split-octonion extension conjecture. Is Theorem 5.1 still true for $D=O$, namely, does every additive map satisfying

$$f(x)=-x^2f(x^{-1})$$

for every $x\in O^\times$ necessarily satisfy $f(x)=0$ for all $x\in O$? The question concerns whether the corresponding result extends from noncommutative alternative division rings to split-octonion algebras; its resolution status is not specified in the supplied text.