Local Ohno–Nakagawa reflection conjecture for symmetric-matrix boxes

Let $K$ be a local field with $\operatorname{char}K\neq2$, let $n$ be odd, and let $\tau$ be a nonzero parameter. Let $\mathcal V_{\tau,\tau^{n-1}f}$ and $\mathcal V_{2\tau^{-1},(2\tau^{-1})^{n-1}f}$ be the indicated integral forms for pairs of symmetric $n\times n$ matrices, and let $g_{\tau,f}:H^1(K,M_f)\mathop{\rightarrow}\limits\mathbb N$ be the associated local orbit counter for a binary $n$-ic form $f$. Local Ohno–Nakagawa reflection conjecture for symmetric-matrix boxes. These two integral forms are naturally dual with duality constant $q^{(n-1)v_K(\tau)}$, and

$$\widehat g_{\tau,f}=|\mathcal O_K/\tau\mathcal O_K|^{n-1}g_{2\tau^{-1},4f}.$$

This is the proposed local reflection theorem underlying the general odd-dimensional box construction. The paper says that, if true, it yields the corresponding global reflection result; no proof is given.