The local-density kernel conjecture for Jordan components of quadratic lattices

Let $(L,h)$ be a quadratic lattice over a non-Archimedean local field, let $\underline{G}$ be the associated smooth integral model, and let $\tilde{G}$ be its special fiber over the residue field $\kappa$. For a Jordan splitting $L=\bigoplus_{i\geq 0}L_i$, let $\bar{V_i}=B_i/Z_i$ be the associated $\kappa$-vector space with nonsingular quadratic form $\bar{h_i}$, and consider the homomorphism

$$\varphi_{\kappa}:\tilde{G}(\kappa)\longrightarrow\prod_i\mathrm{O}(\bar{V_i},\bar{h_i})(\kappa).$$

Local-density kernel conjecture. The map $\varphi_{\kappa}$ is surjective, and its kernel is isomorphic as a set to

$$(\mathbf{A}^{l}\times(\mathbb{Z}/2\mathbb{Z})^{\beta})(\kappa),$$

where

$$l=\dim\tilde{G}-\sum_i\dim\mathrm{O}(\bar{V_i},\bar{h_i})$$

and $\beta$ is a certain non-negative integer.

If true, this gives the remaining group-counting ingredient needed to compute the local density from the smooth integral model. The source presents the claim both as Conjecture 5.10 and as an expectation, but supplies no resolution evidence; its status is therefore open.