In the introduction to his Asterisque Expose "Le Corps des Periodes p-Adiques", Fontaine announces a characterization of $B_{dR}^+$ by some universal property. Unfortunatly, at least for $B_{dR}^+$ itself, this universal property is never spelled out.
Let $K$ denote a finite extension of $\mathbb{Q}_p$, $\mathbb{C}_K$ the completion of its algebraic closure and $\mathcal{O}_K$, $\mathcal{O}_{\mathbb{C}_K}$ the corresponding rings of integers. In section 1.1 Fontaine defines universal objects for various categories: pro-infinitesimal $\mathcal O_K$-thickenings, pro-infinitesial formal $p$-adic thickenings, and infinitesimal thickenings of both kinds up to some order $m \in \mathbb{N}$. In sections 1.2 and 1.3 he explains the construction of the pro-infinitesimal formal $p$-adic $\mathcal O_K$-thickening of $\mathcal O_{\mathbb{C}_K}$ and of the corresponding infinitesimal thickening of order $\leq m$, which he denotes by $A_{inf}$ and $A_{inf}^m$, respectively. Let $A_{inf,K} = A_{inf}[1/p]$ and $A^{m}_{inf,K} = A_{inf}^m[1/p]$, and let $\theta_K : A_{inf,K} \rightarrow \mathbb{C}_K$ denote the map obtained by extending the structure map $\theta : A_{inf} \rightarrow \mathcal{O}_{\mathbb{C}_K}$ of the thickening. Then according to Fontaine, $B^+_{dR}$ is the completion of $A_{inf,K}$ with respect to the $\mathrm{ker}(\theta_K)$-adic topology.
My question: Is $B^+_{dR}$ the universal pro-infinitesimal $\mathcal O_K$-thickening of $\mathbb{C}_K$ ? (Here I mean the first version, not the p-adic formal one.)
As far as I checked, one could prove this by first showing that $A_{inf,K}^m$ is the universal infinitesimal $\mathcal O_K$-thickening of $\mathcal{C}_K$ of order $\leq m$. This seems to be easily done by using the universal property of $A_{inf}^m$. Now the result follows since the pro-infinitesimal thickening is the projective limit over the infinitesimal (finite order) thickenings. This is not explicitly mentioned in the article, but it seems to be obvious from the definitions.
Is this correct, or did I make a serious mistake somewhere? Does someone know a repference where $B^+_{dR}$ as a universal object is addressed? I browsed through all recent accounts on p-adic Hodge theory (Conrad/Brinon, Fontaine/Ouyang, Berger...) but nowhere this issue was discussed.
