Let $X$ a curve on a field $k=\bar{k}$. G a connected reductive group over $k$.

Let fix $d$ closed points $(x_{1},...,x_{d})$ of $X$.

On each point, we have an évaluation morphisme $ev_{x}:G(k[[t_{x}]])\rightarrow G$ and for a Borel $B$ of $G$, we note $I_{x}:=ev^{-1}(B)$.

My question is , is there a choice of Iwahoris $I_{x_{1}}$,..., $I_{x_{d}}$ such that if we have $k_{d}\in G(X-x_{d})$ then there exists a point $k_{1}\in G(X-\{x_{1},\dots,x_{d-1}\})\cap I_{x_{d}}$ such that:

$k_{1}k_{d+1}\in I_{x_{1}}\cap\dots \cap I_{x_{d}}$?