Ohno–Nakagawa duality conjecture for symmetric $2\times n\times n$ boxes

Let $K$ be a local field of characteristic different from $2$, let $n$ be odd, and let $\tau$ be a divisor of $2$. For a binary $n$-ic form $f$, let $\mathcal V_{\tau,\tau^{n-1}f}$ denote the corresponding lattice of symmetric boxes and let $g_{\tau,f}:H^1(K,M)\mathop{\rightarrow}\limits\mathbb N$ be the local orbit counter sending $\alpha$ to the number of self-balanced ideals $J\subseteq R_{f,K}$ representing it.

Local Ohno–Nakagawa box conjecture. The lattices $\mathcal V_{\tau,\tau^{n-1}f}$ and $\mathcal V_{2\tau^{-1},(2\tau^{-1})^{n-1}f}$ are naturally dual with duality constant $q^{(n-1)v_K(\tau)}$. Equivalently,

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

This is the local reflection statement underlying the proposed global Ohno–Nakagawa theorem for odd $n$; it remains conjectural in the paper.