The $(d+1)$-rigidity conjecture for flag spheres

Let $G$ be the graph of a flag triangulation of the $(d-1)$-dimensional sphere. A graph is $(d+1)$-rigid if it has the corresponding generic rigidity property. The $(d+1)$-rigidity conjecture. For all $d\ge 5$, the graph of every flag $(d-1)$-sphere is $(d+1)$-rigid.

If true, this would imply $f_1\ge (d+1)f_0-\binom{d+2}{2}$ for flag spheres of dimension $d-1\ge 4$. The Cone and Gluing Lemmas reduce the conjecture to the case $d=5$, while the assertion is false for $d<5$.