Tensor-stabilizer characterization for forests

Let $G$ be a forest on vertex set $V=\\{1,\ldots,d\\}$, with each connected component a tree having at least three nodes, and let $A$ be its adjacency matrix. Let $\mathcal{V}_G$ be the variety of symmetric tensors whose entries vanish whenever the subgraph induced by their indices is disconnected. Denote by $O(d)$ the orthogonal group, by $SP(d)$ the signed permutation matrices, and by $\mathcal{P}(d)$ the permutation matrices. For $Q\in O(d)$, let $\mathcal{G}_T(\mathcal{V}_G)$ denote the tensor symmetry group of $\mathcal{V}_G$. Forest stabilizer conjecture. Let $Q\in O(d)$. Then $Q\in\mathcal{G}_T(\mathcal{V}_G)$ if and only if $Q\in SP(d)$ and $P^TAP=A$, where $P\in\mathcal{P}(d)$ is a permutation matrix such that $P_{ij}=|Q_{ij}|$. This would characterize all orthogonal transformations preserving the tensor zero structure induced by such a forest.