The $P$-ordinary classicality conjecture for scalar-valued cusp forms

Let $\kappa$ be a very regular dominant weight, let $\psi_B$ be the character occurring in the definition of $\mathcal{V}_{\kappa_p}(K_r,\psi_B;R)$, and let $\Omega_{\kappa,r}$ be the map from classical forms to $p$-adic forms. Write $S^{P\text{-ord}}_\kappa(K_r,\psi_B;R)$ for the $P$-ordinary cusp forms and $\mathcal{V}^{P\text{-ord},\mathrm{cusp}}_{\kappa_p}(K_r,\psi_B;R)$ for the corresponding $P$-ordinary cuspidal $p$-adic forms.

Scalar $P$-ordinary classicality conjecture. The restriction

$$\Omega^{P\text{-ord}}_{\kappa,r}:S^{P\text{-ord}}_\kappa(K_r,\psi_B;R)\longrightarrow\mathcal{V}^{P\text{-ord},\mathrm{cusp}}_{\kappa_p}(K_r,\psi_B;R)$$

of $\Omega_{\kappa,r}$ is an isomorphism.

The statement is known when $P=B$ and is used in the paper to compare $P$-ordinary classical and $p$-adic Eisenstein series.