Pilaud–Santos conjecture on multit triangulations and rigidity matroids

Let $k$-triangulations be the maximal graphs on $[n]$ with no $(k+1)$-crossing, and let the generic bar-and-joint rigidity matroid of dimension $2k$ be the matroid obtained from generic configurations of points in $\mathbb{R}^{2k}$ using the usual bar-and-joint rigidity matrix. Pilaud–Santos conjecture. $k$-triangulations are bases in the generic bar-and-joint rigidity matroid of dimension $2k$. The source states this as a conjecture of Pilaud and Santos; its relationship with the generic hyperconnectivity matroid is given, but the conjecture's resolution is not specified.