Decomposition criterion for equivalence of spatial graphs

Let $\Gamma$ and $\Gamma'$ be finite spatial graphs corresponding to the same combinatorial graph $\mathcal{G}$. A forest–tangle decomposition is a decomposition of a spatial graph as a connected sum

$$\Gamma=\overline{T}\mathbin{\sharp}\overline{\beta},$$

where $\overline{T}$ is a forest and $\overline{\beta}$ is a tangle.

Decomposition criterion. The graphs $\Gamma$ and $\Gamma'$ are equivalent if and only if they admit decompositions

$$\Gamma=\overline{T}\mathbin{\sharp}\overline{\beta},\qquad \Gamma'=\overline{T'}\mathbin{\sharp}\overline{\beta'},$$

where $\overline{T}$ is equivalent to $\overline{T'}$ as spatial graphs and $\overline{\beta}$ is equivalent to $\overline{\beta'}$ as tangles.

This criterion would separate equivalence of spatial graphs into the equivalence of their forest and tangle components, addressing the nonuniqueness of the decomposition arising from the choice of a maximal forest. The supplied text does not establish the claim or indicate whether it has been proved or disproved.