Flawlessness of $h$-vectors of broken circuit complexes

One of the major open questions in matroid theory asks whether the $h$-vector $(h_0,h_1,\dots ,h_s)$ of the broken circuit complex of a matroid $M$ satisfies the following inequalities: $$h_0\le h_1\le \cdots \le h_{\lfloor s\mathrm{/}2\rfloor }\text{and}\text{qua}{h}_i\le h_{s-i}\text{ for }0\le i\le \lfloor s\mathrm{/}2\rfloor \mathrm{.}$$ This paper affirmatively answers the question for matroids that are representable over a field of characteristic zero.