  Let $K$ be a finite extension of $\mathbb{Q}_p$. We study the locally
$\mathbb{Q}_p$-analytic representations $\pi$ of $\mathrm{GL}_n(K)$ of integral
weights that appear in spaces of $p$-adic automorphic representations. We
conjecture that the translation of $\pi$ to the singular block has an internal
structure which is compatible with certain algebraic representations of
$\mathrm{GL}_n$, analogously to the mod $p$ local-global compatibility
conjecture of Breuil-Herzig-Hu-Morra-Schraen. We next make some conjectures and
speculations on the wall-crossings of $\pi$. In particular, when $\pi$ is
associated to a two dimensional de Rham Galois representation, we make
conjectures and speculations on the relation between the Hodge filtrations of
$\rho$ and the wall-crossings of $\pi$, which have a flavour of the
Breuil-Strauch conjecture. We collect some results towards the conjectures and
speculations.
