  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.
