  Finite-dimensional signatures of spinodal criticality are notoriously
difficult to come by. The dynamical transition of glass-forming liquids, first
described by mode-coupling theory, is a spinodal instability preempted by
thermally activated processes that also limit how close the instability can be
approached. We combine numerical tools to directly observe vestiges of the
spinodal criticality in finite-dimensional glass formers. We use the swap Monte
Carlo algorithm to efficiently thermalise configurations beyond the
mode-coupling crossover, and analyze their dynamics using a scheme to screen
out activated processes, in spatial dimensions ranging from $d=3$ to $d=9$. We
observe a strong softening of the mean-field square-root singularity in $d=3$
that is progressively restored as $d$ increases above $d=8$, in surprisingly
good agreement with perturbation theory.
