Rubinstein–Weng–Wormald linear-error conjecture for minimum-topology approximate Steiner trees

Let an $\varepsilon$-approximate Steiner tree be a tree whose angles at Steiner points lie in $[2\pi/3-\varepsilon,2\pi/3+\varepsilon]$. For $d\geqslant 2$, let $\overline{{\mathcal A}}_\varepsilon^d(n)$ be the subset of full $\varepsilon$-approximate Steiner trees on $n$ terminals in $\mathbb R^d$ whose terminals have a minimum Steiner tree with the same topology. If

$$\overline{F}_d(\varepsilon,n)=\sup\left\{\frac{L(T)-L(S(T))}{L(S(T))}:T\in\overline{{\mathcal A}}_\varepsilon^d(n)\right\},$$

where $S(T)$ is the shortest tree with the topology of $T$, then Rubinstein–Weng–Wormald's conjecture. For any $d\geqslant 2$ there exist $\varepsilon_0>0$ and $C_d>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$ and $n\in\mathbb N$,

$$\overline{F}_d(\varepsilon,n)<C_d\varepsilon.$$

The conjecture gives the same uniform linear error bound when the prescribed topology is realized by a minimum Steiner tree; the supplied text states that this case remains open.