Fan realizations for some 2-associahedra

A~$k$-associahedron is a simplicial complex whose facets, called~$k$-triangulations, are the inclusion maximal sets of diagonals of a convex polygon where no~$k+1$ diagonals mutually cross. Such complexes are conjectured for about a decade to have realizations as convex polytopes, and therefore as complete simplicial fans. Apart from four one-parameter families including simplices, cyclic polytopes and classical associahedra, only two instances of multiassociahedra have been geometrically realized so far. This paper reports on conjectural realizations for all~$2$-associahedra, obtained by heuristic methods arising from natural geometric intuition on subword complexes. Experiments certify that we obtain fan realizations of~$2$-associahedra of an~$n$-gon for~$n\in\{10,11,12,13\}$, further ones being out of our computational reach.