The possibility of the existence of a gas-liquid third order phase transition for fluids is becoming a subject of growing interest. Experimental work suggests its existence for specific systems while recent theoretical models claim its universality. In this work, we employ Molecular Dynamics and investigate the third-order phase transition beyond the Andrews critical point by treating a system of Lennard-Jones particles along three isotherms. Two partial derivatives of the Gibbs free energy are measured, namely the molar constant pressureheat capacity and isothermal compressibility. The convergence of these simulations with respect to the system size as well as the cut-off radius is carefully checked. The obtained results show that partial derivatives certainly do not present sharp cusp singularities at the maxima, and actually suggest that there are no singularities at all. On these basis we then conclude that a third-order phase transition in the considered temperature region: T* ⩾ 1.36 may indeed not exist.