Tackling the low-temperature fate of supercooled liquids is challenging due to the immense timescales involved, which prevent equilibration and lead to the operational glass transition. Relating glassy behaviour to an underlying, thermodynamic phase transition is a long-standing open question in condensed matter physics. Like experiments, computer simulations are limited by the small time window over which a liquid can be equilibrated. Here we address the challenge of low temperature equilibration using trajectory sampling in a system undergoing a nonequilibrium phase transition. This transition occurs in trajectory space between the normal supercooled liquid and a glassy state rich in low-energy geometric motifs. Our results indicate that this transition might become accessible in equilibrium configurational space at a temperature close to the so-called Kauzmann temperature, and provide a possible route to unify dynamical and thermodynamical theories of the glass transition.