Highly connected infinite digraphs without edge-disjoint back and forth paths between a certain vertex pair

We construct for all $k\in \mathbb{N}$ a $k$-edge-connected digraph $D$ with $s,t\in V(D)$ such that there are no edge-disjoint $s \rightarrow t$ and $t\rightarrow s$ paths. We use in our construction "self-similar" graphs which technique could be useful in other problems as well.