The possibility of the occurrence of a finite-temperature thermodynamic glass transition in a two-component Lennard-Jones system is investigated using a numerical domain-wall scaling approach. Our study is based on the assumption that some degrees of freedom of the system get frozen below a certain temperature and act as a source of quenched disorder for the remaining ''active'' degrees of freedom. Our results rule out the possibility of the occurrence of a finite-temperature phase transition and suggest a zero-temperature transition with correlation length exponent nu approximate to 1.5.