The wiggly complex polytopality conjecture

Let $P$ be any point set in the plane, and let $\mathrm{WC}_P$ denote its wiggly complex. Wiggly complex polytopality conjecture. For any point set $P$ in the plane, the wiggly complex $\mathrm{WC}_P$ is the boundary complex of a simplicial polytope. The conjecture extends the known cases of collinear point sets, giving the wiggly complex studied in the paper, and point sets in general position, giving the pseudodissection complex. Its status beyond these cases is open; the authors mention possible interpretations via dual pseudoline arrangements and extensions to multi wiggly complexes and arbitrary finite Coxeter groups.