Counterexamples to the Mu--Welker recursive decomposition in every degree

The well-known open problem of Bell and Skandera asks whether a monic real-rooted polynomial $f(t)$ with positive integer coefficients is the $f$-polynomial of a simplicial complex. Mu and Welker proved that if the recursive decomposition $f(t)=g(t)+th(t)$ satisfies the corresponding coefficient ine