Let $X/\mathbb{Z}_p$ be a smooth hyperbolic curve and $\pi^{un}_1(X_{\overline{\mathbb{Q}_p}},b)$ denotes the pro-unipotent completion(over $\mathbb{Q}_p$) of the etale fundamental group of $X$ base changed to $\overline{\mathbb{Q}_p}$ based at a point $b\in X(\mathbb{Z}_p)$. Then we have a map from $X(\mathbb{Z}_p)$ to $H^1(Gal(\overline{\mathbb{Q}_p}/\mathbb{Q}_p),\pi^{un}_1(X_{\overline{\mathbb{Q}_p}},b))$ by sending $x\in X(\mathbb{Z}_p)$ to the (cohomology class of) path torsor $P(b,x)$.

My questions is: is this map always injective?

Let $X/\mathbb{Z}_p$ be a smooth hyperbolic curve and $\pi^{un}_1(X_{\overline{\mathbb{Q}}_p},b)$ denotes the pro-unipotent completion  (over $\mathbb{Q}_p$) of the etale fundamental group of $X$ base changed to $\overline{\mathbb{Q}}_p$, based at a point $b\in X(\mathbb{Z}_p)$. 

Then we have a map $$X(\mathbb{Z}_p) \longrightarrow H^1(Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p), \,\pi^{un}_1(X_{\overline{\mathbb{Q}}_p},b)),$$ defined by sending $x\in X(\mathbb{Z}_p)$ to the (cohomology class of) the path torsor $P(b,x)$.

Question. Is this map always injective?