The density conjecture for scalar-valued $P$-adic modular forms

Let $R$ be the coefficient domain, let $K_r$ denote the principal-congruence levels at $p$, let $\Omega_r$ be the direct sum of the maps from classical scalar-valued modular forms to the space $\mathcal{V}(K^p;R)$ of $p$-adic modular forms, and let $\mathcal{S}^{\mathrm{ord}}$ denote the ordinary locus. Then

Density conjecture. The space

$$\left(\varinjlim_r \Omega_r\left(\bigoplus_{\kappa} H^0(\prescript{}{K_r}{\mathcal{S}^{\mathrm{ord}}}_{/R},\omega_{\kappa,r})\right)\right)[1/p]\cap\mathcal{V}(K^p;R)$$

is $p$-adically dense in $\mathcal{V}(K^p;R)$.

When $P=B$, this is known; the conjecture concerns the corresponding density statement for general $P$-ordinary forms.