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

The well-known open problem of Bell and Skandera asks whether a real-rooted polynomial $f(t)$ with positive integer coefficients and constant term one 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 inequality $h_i<g_i$, then this open problem has an affirmative answer. Mu and Welker also conjectured that the real-rootedness of $f(t)$ implies that of $g(t)$ and $h(t)$. We give counterexamples to the conjecture of Mu and Welker for every degree at least three, and prove that the assertion holds in degrees $1$ and $2$. Moreover, each polynomial we construct is the $f$-polynomial of a simplicial complex.