Ohno–Nakagawa reflection conjecture for odd-dimensional symmetric-matrix boxes

Let $n$ be a positive odd integer. Let $f$ be a polynomial of degree $n$ with no multiple roots and only one real root. A $2\times n\times n$ box is a pair $(A,B)$ of $n\times n$ integer symmetric matrices with resolvent $\det(Ax-B)$. Let $h(f)$ be the number of such boxes with resolvent $f$, up to equivalence, weighted by the reciprocal of the number of symmetries, and let $h_2(f)$ count those whose diagonal entries in both $A$ and $B$ are even, with the same weighting. Ohno–Nakagawa reflection conjecture for $2\times n\times n$ boxes.

$$h_2(2^{n-1}f)=2^{\frac{n-1}{2}}h(f).$$

The claim extends the Ohno–Nakagawa reflection phenomenon from cubic and quartic parametrizations to symmetric-matrix boxes of every positive odd size. The paper states that the $n=3$ case is nearly proved, while larger cases are far from proved.