Arbitrary-position rigidity conjecture for bipartized 2-triangulations

Let a reduced bipartized $2$-triangulation be the bipartite graph associated with a reduced $2$-triangulation, and let the vertices be placed at arbitrary distinct points along the parabola.

Arbitrary-position rigidity conjecture. Reduced bipartized $2$-triangulations are bases in the bipartite bar-and-joint rigidity matroid of these points.

The generic moment-curve result is known in dimension $2$, and experiments suggest that genericity can be replaced by arbitrary distinct positions; the conjecture asserts this stronger statement.