Definition The 2-section of a hypergraph $H$, denoted by $[H]_2$, is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge.
A hypergraph is called eulerian if it admits an euler tour which visits every hyperedge exactly once (Note that euler tour of a hypergraph starts and ends in the same hyperedge but maybe in two different vertices -of the same hyperedge-).
Example The euler tour of the first hypergraph is $\{E_4, E_1,E_2,E_3,E_5,E_4\}$. Note that the euler tour in this case could begin in vertex $v_2$ and ends in vertex $v_3$ (that is euler tour in hypergraph my starts and ends in different vertices -unlike the graph case-).
The 2-section of a hypergraph is illustrated in the second figure
Question My question is if the 2-section graph of the hypergraph $[H]_2$ is eulerian, does this implies that the hypergraph $H$ is also eulerian ?
Comment I'm pretty sure that the above question has positive answer (at least for the case of $k$-uniform hypergraph that is all the hyperedges have the same size $k$), but I do not have any idea how to start!
Any idea will be useful!