TITLE: A curious generalization of Helly's theorem
QUESTION [11 upvotes]: Here is a curious conjectural extension of Helly's theorem.
It may follow (if true) from a useful theorem of the kind asked in this MO question:

Conjecture: Let ${\cal F}=P_1,P_2,\dots,P_m$ be a family all
  whose members are disjoint union of
  two convex sets in $R^d$. Suppose also
  that 
(1) $m \ge d+2$
(2) Every intersection of $i$
  members of $\cal F$, $i < m$ is also
  the disjoint union of two NONEMPTY compact convex sets.
Then the intersections of all members
  of $\cal F$ is not empty.

Remark: Micha A. Perles showed (in the 70s) that even when $d=2$ you cannot replace "two" by "48".

REPLY [2 votes]: This answer shows that you cannot strengthen the conditions of the questions and demand condition (2) only for $m-1$ sets.
Answer for new version IF we only consider the intersection of m-1 sets (as construction fails otherwise, as pointed out by Gil in the comment):
Now it is not hard to prove that the statement is true for d=1 but there is a counterexample for d=2. Take a circle and divide its perimeter into six equal parts, A, B, .., F such that e.g. A, B and C are on the top. Now take a point somewhere high, P, and another somewhere low, Q. Our four sets will be the following: conv(A,P) $\cup$ conv(D,Q), conv(B,P) $\cup$ conv(E,Q), conv(C,P) $\cup$ conv(F,Q) and finally the last set is the disc. Now the intersection of the first three will be around P and Q, while the intersection of the fourth with any two other sets will be around two disjoint arcs of the perimeter.