In this paper, we investigate the boundedness, invariant interval, semicycle and global attractivity of all positive solutions of the equation x n + 1 = α + γ x n − 1 A + B x n + C x n − 1 , n = 0 , 1 , … , where the parameters α , γ , A , B , C ∈ ( 0 , ∞ ) and the initial conditions y − 1 , y 0 are nonnegative real numbers. We show that if the equation has no prime period-two solutions, then the positive equilibrium of the equation is globally asymptotically stable. Our results solve partially the conjecture proposed by Kulenović and Ladas in their monograph [M.R. Kulenović, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman Hall/CRC, Boca Raton, 2001].