Odd-dimensional antisymmetric solution conjecture for noncommutative determinants

Let $n$ be a positive integer, and let $A$ and $B$ be the matrices occurring in the identity $\operatorname{eq.further.1}$, with parameters $h_1$ and $q_i$. A matrix is antisymmetric when it is equal to the negative of its transpose. Odd-dimensional antisymmetric-solution conjecture. If $n$ is odd and at least one of $A$ and $B$ is antisymmetric, then $\operatorname{eq.further.1}$ holds with

$$q_i=(n-i)h_1.$$

This generalizes the antisymmetric solution observed for $n=3$ and $n=5$, but the paper does not establish it for all odd $n$.