Euclidean Steiner tree conjecture for regular simplicial complexes

Let $r\in(0,1)$ and $\alpha\in(0,1/r-1)$. A regular, unit simplicial complex on $m$ vertices has constituent simplices that are regular and unit. Euclidean Steiner tree conjecture for regular simplicial complexes. There exist constants $s,\beta>0$ and a sufficiently large $M\in\mathbb{Z}^+$ such that, for every $m\geq M$, every such complex satisfies both conditions: if its vertices can be partitioned into the vertices of at most $rm$ unit regular simplices, then the $m$-vertex point configuration has a Euclidean Steiner tree of cost at most $sm$; and if its vertices cannot be partitioned into the vertices of fewer than $(1+\alpha)rm$ unit regular simplices, then the point configuration has no Euclidean Steiner tree of cost less than $(1+\beta)sm$. This is the formal gap statement motivating an APX-hardness strategy for Euclidean Steiner Tree. The source does not state a resolution.