Finiteness conjecture for simple connected ordered matroids with prescribed broken-circuit topology

Let $d,k$ be positive integers. For an ordered matroid $M$, write $\overline{BC_<(M)}$ for its reduced broken circuit complex, and let $h_{d-1}(\overline{BC_<(M)})$ denote its $(d-1)$-st $h$-number. Finiteness conjecture. The number of isomorphism classes of simple connected, rank $d$ ordered matroids $M$ satisfying

$$h_{d-1}(\overline{BC_<(M)})=k$$

is finite. This is a finiteness assertion for ordered matroids whose reduced broken circuit complexes have prescribed topology; the supplied text does not indicate whether it has been proved or disproved.