Seminormal extension of the finite-projectivity result for Hodge–Iwasawa modules

Let $X$ be the space attached to $E\{T_1,\ldots,T_n\}$, and let $A_\infty$ be an abelian Fréchet–Stein algebra satisfying the stated sousperfectoid local hypotheses. For a family of $B$-pairs $M$ over $\mathcal{O}\mathbb{B}_{\mathrm{dR},X,A_\infty}$ in the preceding setting, the functor $\mathbb{D}^\bullet_{\mathrm{dR}}$ is coadmissible and finite projective over $\mathcal{O}_X\widehat{\otimes}A_\infty$.

Seminormal extension conjecture. If the space $X$ is seminormal, then the same coadmissibility and finite-projectivity result holds.

The preceding proposition establishes the result under the stated smoothness assumptions, using the connection and a theorem of Zábrádi. The source does not provide a proof or resolution of the seminormal extension.