Spanning even trees of graphs

A tree $T$ is said to be even if all pairs of vertices of degree one in $T$ are joined by a path of even length. We conjecture that every $r$-regular nonbipartite connected graph $G$ has a spanning even tree and verify this conjecture when $G$ has a 2-factor. Well-known results of Petersen and Hanson et al. imply that the only remaining unsolved case is when $r$ is odd and $G$ has at least $r$ bridges. We investigate this case further and propose some related conjectures.