Let $V$ consist of the $n$ choose 2 segments among the vertices of a convex $n$-gon. A subset of $V$ is called $(k+1)$-crossing-free if it does not contain a set of $k+1$ mutually crossing diagonals. Such free subsets form a simplicial complex that we call the $k$-associahedron, since the usual associahedron is (essentially) the case $k=1$. For arbitrary $k$ and $n\ge 2k$ this complex is known to be a shellable sphere of dimension $k(n−2k−1)$, and is conjectured to be polytopal (Jonsson 2003). Since the dimension of the $k$-associahedron happens to coincide with that of any abstract rigidity matrix of dimension $2k$ on $n$ elements, Pilaud and Santos (2010) conjectured that the facets in the complex, called $k$-triangulations, are generically isostatic graphs in dimension $2k$ for bar-and-joint rigidity. This conjecture was intended as a step towards the construction of the $k$-associahedron as a polytope (or at least as a complete fan) using as facet normals the rows of the corresponding rigidity matroid in a suitable embedding. Together with Luis Crespo I have explored this possibility in recent years and we have found that: A. Restricted to the moment curve, bar-and-joint rigidity coincides with another two classical forms of rigidity, Whiteley’s cofactor rigidity and Kalai’s hyperconnectivity. The former is particularly suitable to model k-triangulations since it is based on configurations of points in the plane. The latter connects rigidity and $k$-triangulations with the variety of Pfaffians of antisymmetric matrices of size $2k$, which allows us to prove that $k$-triangulations are indeed generically isostatic with respect to hypercopnnectivity, B. Suitable choices of points along the moment curve allow us to find realisations of the $k$-associahedron, or at least its fan, for all values of $(k,n)$ with $n≤13$, except for $(3,12)$ and $(3,13)$. C. We prove that no choice of points along the moment curve can provide realisations of the $k$-associahedron for any n≥2k+6 and k≥3. This includes the cases (3,12) and (3,13) mentioned above.