  In [BS07] Breuil and Schneider formulated a conjecture on the equivalence of
the existence of invariant norms on certain $p$-adically locally algebraic
representations of $GL_n(F)$ and the existence of certain de-Rham
representations of $Gal(\bar{F}/F)$, where $F$ is a finite extension of
$\mathbb{Q}_p$. In [Bre03b, DI13] Breuil and de Ieso proved that in the case $n
= 2$ and under some restrictions, the existence of certain admissible
filtrations on the $\phi$-module associated to the two-dimensional de-Rham
representation of $Gal(\bar{F}/F)$ implies the existence of invariant norms on
the corresponding locally algebraic representation of $GL_2(F)$. In [Bre03b,
DI13], there is a significant restriction on the weight - it must be small
enough. In [CEG+13] the conjecture is proved in greater generality, but the
weights are still restricted to the extended Fontaine-Laffaille range. In this
paper we prove that in the case $n = 2$, even with larger weights, under some
restrictions, the existence of certain admissible filtrations implies the
existence of invariant norms.
