Connected even factors in the square of essentially 2-edge-connected graph

An essentially k-edge connected graph G is a connected graph such that deleting less than k edges from G cannot result in two nontrivial components. In this paper we prove that if an essentially 2-edge-connected graph G satisfies that for any pair of leaves at distance 4 in G there exists another leaf of G that has distance 2 to one of them, then the square G² has a connected even factor with maximum degree at most 4. Moreover we show that, in general, the square of essentially 2-edge-connected graph does not contain a connected even factor with bounded maximum degree.