Wild-kernel annihilation conjecture

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$, let $r>1$ be an integer, let $p\ne2$ be a prime, and let $S$ contain $S_{\mathrm{ram}}\cup S_\infty\cup S_p$. Assume Schneider's conjecture and Gross's conjecture in the form used in the paper. Let $\mathcal J_r^S$ be the canonical fractional Galois ideal and let $\mathcal H_p(G)$ be the relevant denominator ideal. Wild-kernel annihilation conjecture. For every $x\in\operatorname{Ann}_{\mathbb Z_p[G]}(\mathbb Z_p(r-1)_{G_L})$, one has

$$\operatorname{Nrd}_{\mathbb Q_p[G]}(x)\,\mathcal H_p(G)\,\mathcal J_r^S\subseteq\operatorname{Ann}_{\mathbb Z_p[G]}(K_{2r-2}^w(\mathcal O_{L,S})_p).$$

This is the paper's non-abelian annihilation statement for Banaszak's $p$-adic wild kernel, conditional on the stated Gross and Schneider conjectures; its resolution is not specified.