The non-vanishing conjecture for $B_{\xi,\zeta}$

Let $G$ be a split connected reductive group over $L$, let $\rho_U$ be the algebraic representation associated with a highest weight $\xi$, and let $\zeta\in\mathbf{T}'_{\xi,\mathrm{norm}}$ be a point in the norm affinoid of the dual torus. Form the Banach representation

$$B_{\xi,\zeta}:=K_\zeta\widehat{\otimes}_{\mathcal{B}(G,\rho_U)}B_U^G(\rho_U).$$

Non-vanishing conjecture for $B_{\xi,\zeta}$. The Banach space $B_{\xi,\zeta}$ is non-zero. This extends the preceding Satake and unramified Langlands-functoriality construction to locally algebraic Banach representations; the supplied text gives no resolution.