Summary: We consider the second order rational difference equation

${x}_{n+1}=(\alpha +\beta {x}_{n}+\gamma {x}_{n-1})/(A+B{x}_{n}+C{x}_{n-1}),n=0,1,2,\cdots $, where the parameters

$\alpha ,\beta ,\gamma ,A,B,C$ are positive real numbers, and the initial conditions

${x}_{-1},{x}_{0}$ are nonnegative real numbers. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable. In particular, we solve Conjecture 5.201.2 proposed by

*E. Camouzis* and

*G. Ladas* [Dynamics of third-order rational difference equations with open problems and conjectures. Advances in Discrete Mathematics and its Applications 5. Boca Raton, FL: Chapman & Hall/CRC (2008;

Zbl 1129.39002)] which appeared previously in Conjecture 11.4.3 in [

*M. R. S. Kulenović* and

*G. Ladas*, Dynamics of second order rational difference equations. With open problems and conjectures. Boca Raton, FL: Chapman & Hall/CRC (2002;

Zbl 0981.39011)].