Intersection patterns of planar sets

Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $S$ of $\{1,\ldots,n\}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in S} A_i \neq \emptyset$. Let $k \geq 2, b\geq 1$ be integers. We prove that if all $k$-wise and $(k+1)$-wise intersections of $\mathcal A$ are open and have at most $b$ path-connected components, then $f_{k+1}=0$ implies $f_k \leq cf_{k-1}$ for some positive constant $c$ depending only on $b$ and $k$. The result also extends to two-dimensinal compact connected surfaces.