The feasible ideal-point conjecture for the MIN NODES–MIN DOUBLE problem

Let $\mathcal{V}$ be the vertex set of a discretizable distance geometry problem, let $rank$ assign vertices to ranks, and let $nodes(rank(v))$ denote the number of nodes at the rank of $v$. Let $double(rank^{-1}(r))$ indicate whether the vertex at rank $r$ is a double, and consider the multi-objective problem

$$\min \left\{\sum_{v\in\mathcal{V}} nodes(rank(v)),\sum_{r\in[n-1]} double(rank^{-1}(r))\right\}$$

subject to the rank and predecessor constraints. Feasible ideal-point conjecture. The $\texttt{MIN NODES}$–$\texttt{MIN DOUBLE}$ multi-objective problem has a feasible ideal point; equivalently, its Pareto frontier consists of a single non-dominated point. If true, solving $\texttt{MIN DOUBLE}$ would yield an optimal solution to $\texttt{MIN NODES}$. The conjecture concerns simultaneous optimization of the number of nodes and doubles in discretizable distance geometry problems; the supplied text gives no resolution status.