Towards the wall-crossing of locally Qₚ-analytic representations of GLₙ(K) for a p-adic field K

Let K be a finite extension of Qₚ. We study the locally Qₚ-analytic representations π of GLₙ(K) of integral weights that appear in spaces of p-adic automorphic representations. We conjecture that the translation of π to the singular block has an internal structure which is compatible with certain algebraic representations of GLₙ, 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 π. In particular, when π is associated to a two dimensional de Rham Galois representation, we make conjectures and speculations on the relation between the Hodge filtrations of ρ and the wall-crossings of π, which have a flavour of the Breuil-Strauch conjecture. We collect some results towards the conjectures and speculations.