The generalized Steinberg invariant norm conjecture

Let $G$ be the group in the paper, let $\rho$ be an irreducible algebraic $\mathbb{Q}_p$-rational representation of $G$ over $K$, and let $\pi$ be a generalized Steinberg representation of $G$ over $K$. Generalized Steinberg invariant norm conjecture. The representation $\rho\otimes_K\pi$ admits an invariant norm if and only if its central character is integral. This is obtained by combining a proposition on generalized Steinberg representations with the paper's main conjecture; no resolution is specified here.