Weighted 3-Wise 2-Intersecting Families

Let n and r be positive integers. Suppose that a family F⊂2[n] satisfies ∣F1∩F2∩F3∣⩾2 for all F1, F2, F3∈F. We prove that if w<0.5018, then ∑F∈Fw∣F∣(1−w)n−∣F∣⩽w2.