The associahedron has a natural extension called the multiassociahedron. It is a vertex-decomposable simplicial sphere (in other words, a "combinatorially nice" sphere). Since at least 2003, people tried to construct simplicial polytopes, whose boundary is this simplicial complex, with limited success. In this talk, I will reveal (at least) three combinatorial "known-to-be-challenging" computational problems which should be indispensably faced to solve this problem.