Let $\Gamma$ be a finite, connected graph (multiple edges between two vertices are allowed). Fix a vertex $u_0\in V\Gamma$.
Does there exist a maximal subtree (i.e., a spanning tree) $T\subset\Gamma$ such that $ |d_T(\alpha(e),u_0)-d_T(\omega(e),u_0)|\leq1 \quad\text{for all edges } e\in E\Gamma\setminus ET? $ (Here $d_T$ is the path metric in $T$, and $\alpha(e)$ and $\omega(e)$ are the initial and terminal vertices.)
If such a subtree does not exist in general, does this at least hold for planar graphs $\Gamma$?
I have tried induction arguments, but the details don't seem to work out. Can someone give me a hint as to what to try?