Polynomial-time fixed-topology minimum-bead tree conjecture

Let $N$ be a set of terminals embedded in the Euclidean plane, and let $G$ be any tree topology on $N$ whose Steiner points have degree at least three. Fixed-topology bead-minimization conjecture. Finding a tree $T$ on $N$ with the same topology as $G$ and minimizing $\mathrm{beads}(T)$ can be done in polynomial time. This conjecture is motivated by the prospect of a polynomial-time algorithm based on displacing Steiner points of a Steiner minimum tree; the supplied text gives no resolution.