Extension conjecture for locally analytic functions on the bc-ordinary Igusa variety

Let

$$\mathscr{N}^{\dagger}:=\varinjlim^{\mathrm{lcv}}_U \operatorname{H}^0(U,\mathcal{O}_{N\backslash P^{\operatorname{an}}_{\operatorname{dR}}}),$$

where the locally convex inductive limit is over open neighborhoods $U\subset N\backslash P^{\operatorname{an}}_{\operatorname{dR}}$ containing the closure of the image of $\mathfrak{Ig}_{\mathrm{M},\eta}$. The algebra $\mathcal{O}^{\gamma\operatorname{-la}}(T_pH^{\vee})$ acts on $\mathcal{O}(\mathfrak{Ig}_{\mathrm{M},\eta})$. Extension conjecture. This action extends to an action of $\mathcal{O}^{\gamma\operatorname{-la}}(T_pH^{\vee})$ on $\mathscr{N}^{\dagger}$. This is proposed as a $\mu$-ordinary generalization of the preceding theorem on the action of locally analytic functions. The supplied text gives no resolution of the conjecture.