Let $\mathbf{f}$ be a Hida family of ordinary $p$-adic modular forms, and let $V(\mathbf{f})$ be the corresponding $\Lambda$-adic Galois representation (a quotient of the inverse limit $$ \varprojlim_r H^1_{et}(X_1(Np^r), \mathbf{Z}_p) $$ with respect to the trace maps).

It follows from work of Wiles that if the residual representation of $\mathbf{f}$ is irreducible and $p$-distinguished, then $V(\mathbf{f})$ is free of rank 2 over the Hida--Hecke algebra $\Lambda_{\mathbf{f}}$ and there is a canonical exact sequence $$ 0 \to \mathscr{F}^+ V(\mathbf{f}) \to V(\mathbf{f}) \to \mathscr{F}^- V(\mathbf{f}) \to 0 \quad (\star)$$ where the filtration steps are free of rank 1 over $\Lambda_{\mathbf{f}}$ and $\mathscr{F}^- V(\mathbf{f})$ is unramified.

Ohta has shown (Compos. Math. 115, 1999) that if the weight of $\mathbf{f}$ is not 2 modulo $p-1$, then there is a canonical basis of the module $$ \mathbf{D}_{\mathrm{dR}}(\mathscr{F}^- V(\mathbf{f})) := \left( \mathscr{F}^- V(\mathbf{f}) \mathop{\hat\otimes} \widehat{\mathbf{Z}}_p^{\mathrm{nr}}\right)^{G_{\mathbf{Q}_p}}$$ which interpolates the images of the normalized differentials $\omega_{f^*}$ under the de Rham comparison isomorphisms for each classical specialization $f$ of $\mathbf{f}$. (Ohta also assumes the weight is not $1$ modulo $p-1$; but from his method is it clear that this can be weakened to assuming p-distinguishedness.)

(1) Can the assumption in Ohta's theorem that the weight is not 2 modulo $p-1$ be removed?

(2) The exact sequence $(\star)$ exists for arbitrary $\mathbf{f}$ (not necessarily p-distinguished) if we tensor with the field of fractions of the Hecke algebra, by a theorem of Hida and Wiles. Can Ohta's $p$-adic interpolation of periods be generalized to this setting?