A proof of the Hoggatt-Bergum conjecture

It is proved that if k and d are positive integers such that the product of any two distinct elements of the set {F2k, F2k+2, F2k+4, d} increased by 1 is a perfect square, than d has to be 4F2k+1F2k+2F2k+3. This is a generalization of the theorem of Baker and Davenport for k = 1.