Local Ohno–Nakagawa duality conjecture for traced quartic rings

Let $K$ be a $2$-adic local field and let $0\leq t\leq s\leq e=v_K(2)$. Let $C_1\subseteq C_2$ be orders in a cubic $K$-algebra $R$ with $C_2/C_1\cong\mathcal{O}_K/\pi^{s-t}$ cyclic as an $\mathcal{O}_K$-module. For each quartic $K$-algebra $L$ with resolvent $R$, let $g(L,C_1,C_2,t,s)$ count $(t,s)$-traced quartic rings in $L$ with reduced resolvent and coresolvent $C_1,C_2$. Local Ohno–Nakagawa duality conjecture for traced quartic rings.

$$\widehat g(L,C_1,C_2,t,s)=q^{t+s}g(L,C_1,C_2,e-s,e-t),$$

where $q=|k_K|$. This conjectural local self-duality would supply the local input for a global reflection theorem; when $t+s=e$, it asserts self-duality up to the stated scaling factor. The paper presents it as unproved.