Strong Erdős–Frankl–Füredi lower-bound conjecture

Prove that the threshold T(r), defined as the least integer n for which an n-element ground set admits an r-cover-free family with more than n members, satisfies T(r)\ge (r+1)^2 for every integer r\ge1.

Background

The paper defines T(r)=\min{n:f(n,r)>n}, where f(n,r) is the maximum size of an r-cover-free family on an n-element ground set. Its main theorem establishes the asymptotic Erdős–Frankl–Füredi conjecture by proving the lower bound T(r)\ge r^2+r+2, which is sufficient to show T(r)=(1+o(1))r^2.

The authors note that Erdős, Frankl, and Füredi proposed the stronger bound T(r)\ge(r+1)^2, but explicitly state that their result does not settle it. Combined with the affine-plane upper-bound construction, this stronger conjecture would imply the exact value T(r)=(r+1)^2 whenever r+1 is a prime power, thereby establishing the optimality of that construction.