Convex-polytope f-vector unimodality conjecture

Let $P$ be a convex $d$-polytope with $f$-vector

$$f(P)=(f_0,f_1,\ldots,f_{d-1}),$$

where $f_k$ is the number of $k$-dimensional faces of $P$. The unimodality conjecture. For each $d$-polytope $P$ there is an integer $\ell=\ell(P)$ such that

$$f_0\le f_1\le\ \cdots\ \le f_{\ell}\ge\ \cdots\ \ge f_{d-2}\ge f_{d-1}.$$

The conjecture was posed at least twice, by Theodore Motzkin in the late 1950s and by Dominic Welsh in 1972, but the source states that it is disproved, apparently by a construction of Ludwig Danzer presented in 1964. It is therefore not an open conjecture.