Component spanning trees avoiding any prescribed label

Let $\hat{\mathcal{G}}_{2n+1}$ be the graph whose components $A$ have labeled edges, and let $\eta$ be a label of a component $A$. A component spanning tree of $A$ is a spanning tree using vertices of $A$ and edges of the component. Label-avoiding spanning-tree conjecture. For each component $A$ in $\hat{\mathcal{G}}_{2n+1}$, with $n\geq 3$, and for each label $\eta$ of $A$, there exists a component spanning tree of $A$ containing no edge with label $\eta$. This assertion would provide the nearly spanning trees needed by the proposed snake construction, but the source does not report a proof for general $n$.