The rigidity conjecture for multiassociahedra along the moment curve

Let $P$ be a convex $n$-gon, and let $k$-triangulations be its maximal sets of diagonals containing no $k+1$ mutually crossing diagonals. Place $n$ generic points on the moment curve $\{(t,t^2,\dots,t^{2k})\in\mathbb{R}^{2k}:t\in\mathbb{R}\}$. The associated bar-and-joint rigidity matroid is the matroid represented by the rows of the rigidity matrix of these points. The rigidity conjecture. Every $k$-triangulation of the $n$-gon is isostatic, that is, a basis of this bar-and-joint rigidity matroid. The conjecture would imply the corresponding statement for generic points, as well as for the generic cofactor-rigidity and hyperconnectivity matroids; the hyperconnectivity case is known, and the case $k=2$ is proved for generic positions along the moment curve. The conjecture remains open in general.