The PVMD power-series complete integral-closure conjecture

Let $D$ be a Prüfer $v$-multiplication domain (PVMD), and let $D[[X]]$ denote its formal power-series ring.

Power-series complete integral-closure conjecture. $D[[X]]$ is integrally closed if and only if $D[[X]]$ is completely integrally closed, and this holds if and only if $D$ is completely integrally closed.

The paper presents this as a deliberately tentative conjecture motivated by the fact that integral closedness of $D[[X]]$ implies that $D$ is integrally closed and Archimedean. No example is known in the stated setting separating these properties, but the authors explicitly regard the conjecture as unlikely and hope for counterexamples.