Breuil–Schneider non-vanishing conjecture for $B_{\xi,\zeta}(G)$

Let ${\bf G}$ be an $F$-split connected reductive algebraic group defined over $F$, let ${\bf T}$ be a maximal $F$-split torus, and let $\xi$ and $\zeta$ be the parameters used to define the Banach space $B_{\xi,\zeta}(G)$. Let ${\bf T}^\vee_{\xi,\rm{norm}}$ be the specified affinoid subdomain of the dual torus, and for $\zeta\in{\bf T}^\vee_{\xi,\rm{norm}}$ define $B_{\xi,\zeta}(G)$ by the completed Hecke-algebra tensor product described in the source. Breuil–Schneider non-vanishing conjecture. The Banach space $B_{\xi,\zeta}(G)$ is non-zero for every $\zeta\in{\bf T}^\vee_{\xi,\rm{norm}}$. This is the explicit non-vanishing formulation of the Breuil–Schneider prediction in the setting of the paper. The source gives no resolution status beyond stating the conjecture.