Identities with inverses on matrix rings over a division ring of characteristic two

Let $\mathcal{D}$ be a division ring with char $(\mathcal{D})=2$. Let $\mathcal{R}=M_n(\mathcal{D})$ be the ring of all $n\times n$ matrices over $\mathcal{D}$, where $n\ge 2$. Let $f,g:\mathcal{R}\to \mathcal{R}$ be two additive maps such that

$$f(A)+Ag(A^{-1})A=0$$

for all invertible $A\in \mathcal{R}$. If $|\mathcal{D}|\ne 2,4,8$, then

$$f(A)=AQ+\delta (A)\text{and}g(A)=QA+\delta (A)$$

for all invertible $A\in \mathcal{R}$, where $A\in \mathcal{R}$ and $\delta$ is a derivation of $\mathcal{R}$. This result affirmatively answers a question posed by Argac et al. under a mild condition.