Stoker-type conjecture for infinitesimal dihedral-angle rigidity of polytopes

Let $S(t)$ be a differentiable one-parameter family of oriented piecewise-linear surfaces. For each facet $F$, let $n_F$ be its unit normal and $V_F$ its volume, and define the dihedral angle between adjacent facets $F$ and $G$ by $\theta_{FG}:=\angle(n_F,n_G)$. Dots denote derivatives at $t=0$. Stoker-type conjecture. If $\dot\theta_{FG}=0$ for all facets adjacent at $t=0$, then

$$\sum_i \dot V_i n_i=\sum_i V_i\dot n_i=0.$$

This is the explicit conjectural formulation of the stress-flex reformulation and is equivalent to the preceding statement in the source. The surrounding article resolves the weak stress-flex version for polytopes, while the broader Stoker-type formulation is the claim identified here.