Finite string system conjecture for finitely supported loads

Let $\Omega$ be a bounded convex domain, and let the load $f \in \mathcal{M}(\overline{\Omega};\mathbb{R})$ be finitely supported. An optimal string system consists of a pair $(\hat\Pi,\hat\pi)$ solving the optimal string system problem. Write $\mathrm{sp}(\hat\Pi,\hat\pi)$ for its support and $\mathrm{sp}\,f$ for the support of the load. Finite string system conjecture. A solution of the optimal string system problem exists. Moreover, there is at least one solution $(\hat\Pi,\hat\pi)$ satisfying

$$\mathrm{sp}(\hat\Pi,\hat\pi) \subset (\partial\Omega \cup \mathrm{sp}\,f) \times (\partial\Omega \cup \mathrm{sp}\,f).$$

The conjecture predicts that finitely supported loads admit an optimal finite string system whose strings connect only boundary points or points in the support of the load. Its status is not established by the supplied source context.