Real-part conjecture for tree amplitude functions

Let $T$ be a tree, meaning a connected circuit-free non-oriented finite simple graph, and let $C_T(\mathbb{R}^3)$ be its configuration space. Let $\mathcal{A}_T$ be the associated amplitude function. Tree real-part conjecture. For every $\mathbf{x}\in C_T(\mathbb{R}^3)$,

$$\operatorname{Re}(\mathcal{A}_T(\mathbf{x}))\geq 1.$$

If true, this conjecture implies the graph-amplitude lower-bound conjecture, and hence also the nonvanishing conjecture, for trees; its resolution is not stated in the supplied text.