Grid-shell optimality conjecture for vaults under finitely many point loads

Let $f$ be a finite collection of point forces, written as

$$f = \sum_{j=1}^{N} P_j\,\delta_{x_j}.$$

A truss-like solution of $(\mathcal{P})$ is a measure of the form

$$\sigma= \sum_{k=1}^M s_k\,\sigma^{\,x^-_k,\,x^+_k} = \sum_{k=1}^M s_k\,\tau_k \otimes \tau_k\,\mathcal{H}^1\mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex}[\,x^-_k,\,x^+_k], \qquad \tau_k = \frac{x^+_k-x^-_k}{\left\lvert x^+_k-x^-_k\right\rvert},$$

where $x^-_k,x^+_k$ belong to $\partial\Omega\cup\operatorname{spt}f=\partial\Omega\cup\{x_1,\ldots,x_N\}$. A vault optimally designed for a discrete load is a grid-shell. There exists such a truss-like solution $\sigma$ of $(\mathcal{P})$, and the optimal vault is therefore composed of finitely many bars joining boundary points and points of force application.