Generalized S-genus identity for ternary quadratic forms over totally real fields

Let $F$ be a totally real number field of class number one in which $2$ is inert, and let $A_F$ be its ring of integers. Suppose that $2S(A_F^\times)^2$ is the squarefree determinant squareclass of a totally positive definite $A_F$-valued binary quadratic form, and that $W\bmod A_F$ divides $S$. For a ternary quadratic form $Q$, define

$$\epsilon_W(Q):=\prod_{\mathfrak{p}\mid W}\varepsilon_{\mathfrak{p}}(\operatorname{Gen}(Q)).$$

Let $r_Q(m)$ denote the representation number and $\operatorname{Aut}(Q)$ the automorphism group. Generalized S-genus identity. There is a constant $\kappa_F\in\mathbb{Q}_{>0}$ depending only on $F$ such that, for every totally positive $m\in A_F$ satisfying $W\mid m$ and $m\in(A_F^\times)^2\cup 2(A_F^\times)^2\pmod{4A_F}$,

$$\sum_{Q\in\text{$S$-genus}}\epsilon_W(Q)\frac{r_Q(m)}{|\operatorname{Aut}(Q)|}=\kappa_F W^{[F:\mathbb{Q}]}\frac{r_{x^2+y^2+z^2}(m/W^2)}{|\operatorname{Aut}(x^2+y^2+z^2)|}.$$

This proposes a generalization of the proved S-genus identities from the rational setting to totally real number fields satisfying the stated hypotheses. The constant is asserted to depend only on $F$, while the identity is required uniformly for all admissible totally positive $m$.