Bipartiteness conjecture for graphs with the complete identity property after attaching trees

Let $G$ be a connected graph. A graph has the complete identity property if every vertex can serve as the sink and the recurrent identity is the same configuration. If, for every vertex $v\in G$, attaching a single tree of size $1$ to $v$ results in a graph with the complete identity property, then

Bipartiteness conjecture. $G$ is bipartite.

The conjecture is motivated by a computer search showing that every connected graph with at most $10$ vertices having this property is bipartite, although not every bipartite graph has the property. Its general validity remains open.