Let \(\bar{x}\) be the positive equilibrium of the difference equation \[ x_{n+1}=\frac{\alpha + \beta x_{n-1} + \gamma x_n}{A+ B x_{n-1} +C x_n},\quad n=0,1,2,\dots \] where the parameters, \(\alpha,~\beta,~\gamma,~A,~B,~C\), and the initial conditions, \(x_{-1},~x_0\), are nonnegative such that \(A+B+C>0\), \(\alpha+\beta+\gamma>0\), and the denominator is always positive. The author utilized an inclusion theorem due to \textit{L. Berg} [J. Difference Equ. Appl. 10, 399--408 (2004; Zbl 1056.39003), corrections ibid. 11, No. 2, 181--182 (2005; Zbl 1080.39002)] and proved that under the assumption \(B\bar{x}<\beta\) and \(C\bar{x}\geq \gamma\), the above difference equation possesses solutions with a single semicycle. This result confirms positively Conjectures 4.8.3 and 5.4.6 in the monograph [\textit{M. Kulenović} and \textit{G. Ladas}, Dynamics of second-order rational difference equations. With open problems and conjectures. Boca Raton, FL: Chapman and Hall/CRC (2002; Zbl 0981.39011)].