Bjrner's quarter-monotonicity conjecture for convex-polytope f-vectors

Let $P$ be a convex $d$-polytope with $f$-vector $(f_0,f_1,\ldots,f_{d-1})$, where $f_k$ denotes the number of $k$-dimensional faces. Bjrner's quarter-monotonicity conjecture. The $f$-vector increases strictly through the first quarter and decreases strictly through the last quarter:

$$f_0<f_1<\ \cdots\ <f_{\lceil\frac{d-1}{4}\rceil},\qquad\qquad f_{\lfloor\frac{3(d-1)}{4}\rfloor}>\ \cdots\ >f_{d-2} >f_{d-1}.$$

This is the part of unimodality that remains after the general unimodality conjecture fails. The source says it is trivially true for $d\le5$ and true for simplicial $d$-polytopes, using the necessity part of the $g$-theorem; its status for general convex polytopes is presented as open.