Rigidity realization conjecture for reduced bipartized k-triangulations

Let a $k$-triangulation be a maximal set of diagonals of an $n$-gon containing no $k+1$ pairwise crossing diagonals, and let its reduced bipartization be the associated bipartite graph. Place the relevant vertices at generic points on the moment curve in $\mathbb{R}^{k}$.

Rigidity realization conjecture. Reduced bipartized $k$-triangulations of the $n$-gon are bases in the bar-and-joint rigidity matroid of these generic points.

The preceding theorem establishes the corresponding basis statement for the generic bipartite $k$-hyperconnectivity matroid and independence in the generic $k$-rigidity matroid. The conjecture is stronger and would imply the analogous result for generic bar-and-joint rigidity; the case $k=2$ follows from the stated theorem.