Haensch–Kane spinor-genus decomposition conjecture for ternary lattice cosets

Let $L$ be a quadratic lattice and let $\nu\in\mathbb{Q}L$. Define the proper spinor-genus theta series by

$$\Theta_{\operatorname{spn}^+(L+\nu)}:=\frac{1}{\sum_{K+\mu\in\operatorname{spn}^+(L+\nu)}o^+(K+\mu)^{-1}}\sum_{K+\mu\in\operatorname{spn}^+(L+\nu)}\frac{\Theta_{K+\mu}}{o^+(K+\mu)},$$

where the sum is over representatives of the proper classes in the proper spinor genus of $L+\nu$, and $o^+(K+\mu)$ is the number of proper automorphs of the lattice coset. Haensch–Kane's spinor-genus decomposition conjecture. For every quadratic lattice $L$ and $\nu\in\mathbb{Q}L$,

$$\Theta_{\operatorname{spn}^+(L+\nu)}=E_{L+\nu}+\mathcal{U}_{\operatorname{spn}^+(L+\nu)},$$

where $\mathcal{U}_{\operatorname{spn}^+(L+\nu)}$ is a linear combination of unary theta functions. This extends Schulze-Pillot's corresponding decomposition for lattices to lattice cosets and is motivated by applications to representations of sufficiently large integers. The supplied text does not establish whether the conjecture has been resolved.