Flawlessness conjecture for broken circuit complex h-vectors

Let $M$ be a loopless matroid, and let $(h_0,h_1,\ldots,h_s)$ be the $h$-vector of its broken circuit complex $BC(M)$, where $s$ is the largest index $j$ such that $h_j\ne0$. Flawlessness conjecture. The inequalities

$$h_0\leq h_1\leq \cdots\leq h_{\lfloor s/2\rfloor}$$

and

$$h_i\leq h_{s-i}\qquad\text{for }0\leq i\leq \lfloor s/2\rfloor$$

hold. Characterizing the $f$-vectors and $h$-vectors of broken circuit complexes is widely regarded as out of reach, so restrictions such as these provide a more realistic direction for understanding them.