Strong stress-flex conjecture for coned frameworks over closed PL surfaces

Let $S$ be a closed piecewise-linear surface, and let $(G_S^\star,\boldsymbol p)$ be its coned framework: the framework has the vertices and edges of $S$, together with a cone vertex $p_\star$ joined to every vertex of $S$. A first-order flex is an infinitesimal motion preserving the edge lengths to first order, and a stress $\boldsymbol\omega$ is an equilibrium stress of this framework. Strong stress-flex conjecture. If $\dot{\boldsymbol p}$ is any first-order flex with $\dot p_\star=0$, and $\boldsymbol\omega$ is any stress of $(G_S^\star,\boldsymbol p)$, then

$$\sum_i \omega_{\star i}\dot p_i=0.$$

The paper describes this as a stronger version of the weak stress-flex conjecture, suggested by experiments and expected to hold in substantially greater generality than the polytope case resolved in the article. Its general validity remains open.