Breuil–Schneider conjecture in the generic case

Let $F$ be a finite extension of ${\mathbb Q}_p$, let $E$ be a sufficiently large coefficient field, and write $G=GL_n(F)$. Let $\pi$ be an absolutely irreducible generic representation of $GL_n(F)$ and let $\sigma$ be an irreducible algebraic representation of $\operatorname{Res}_{F/{\mathbb Q}_p}GL_n/F$, both over $E$. A representation admits a $G$-invariant norm when it has a norm invariant under the action of $G$. Breuil–Schneider conjecture in the generic case. The following statements are equivalent:

$$\pi\otimes_E\sigma\text{ admits a }G\text{-invariant norm}.$$

There is a potentially semistable Galois representation $r:G_F\to GL_n(E)$ such that

$$\pi=\pi_{sm}(r),\qquad \sigma=\pi_{alg}(r).$$

This is the generic case of the conjecture relating integral structures on irreducible locally algebraic representations to potentially semistable Galois representations. The implication from the Galois-representation condition to the invariant-norm condition is known in full generality by Hu; the converse is therefore the unresolved direction in the formulation given here.