Hodge structure on the fundamental group and its application to $p$-adic integration

We study the unipotent completion $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)$ of the de Rham fundamental groupoid of a smooth algebraic variety over a local non-Archimedean field $K$ of characteristic 0. We show that the vector space $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)$ carries a certain additional structure. That is a $\mathbb Q^{\rm ur}_p$-space $\Pi_{\rm un}(x_0,x_1,X_K)$ equipped with a $\sigma$-semi-linear operator $\phi$, a linear operator $N$ satisfying the relation $N\phi=p\phi N$, and a weight filtration $W_\cdot$ together with a canonical isomorphism $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)\otimes_K \overline K\simeq\Pi_{\rm un}(x_0,x_1,X_K)\otimes_{\mathbb Q_{\rm p}}^{\rm ur}\overline K$. We prove that an analogue of the monodromy conjecture holds for $\Pi_{\rm un}(x_0,x_1,X_K)$.
As an application, we show that the vector space $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)$ possesses a distinguished element. In other words, given a vector bundle $E$ on $X_K$ together with a unipotent integrable connection, we have a canonical isomorphism $E_{x_ 0}\simeq E_{x_1}$ between the fibres. This construction is a generalisation of Colmez's p-adic integration $({\rm rk}E=2)$ and Coleman's $p$-adic iterated integrals ($X_K$ is a curve with good reduction).
In the second part, we prove that, for a smooth variety $X_{K_0}$ over an unramified extension of $\mathbb Q_p$ with good reduction and $r\leq\frac{p-1}{2}$, there is a canonical isomorphism $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)\otimes B_{\rm dR}\simeq\Pi_{r}^{\rm et}(x_0,x_1,X_{\overline K_0})\otimes B_{\rm dR}$ compatible with the action of the Galois group ($\Pi^{\rm dR}_{\rm r}(x_0,x_1,X_{K_0})$ stands for the level $r$ quotient of $\Pi^{\rm dR}_{\rm un}(x_0,x_1,X_K)$). In particular, this implies the crystalline conjecture for the fundamental group (for $r\leq\frac{p-1}{2}$).