  We investigate the slow dynamics of a colloidal model with two repulsive
length scales, whose interaction potential is the sum of a hard-core and a
square shoulder. Despite the simplicity of the interactions, Mode-Coupling
theory predicts a complex dynamic scenario: a fluid-glass line with two
reentrances and a glass-glass line ending with multiple higher-order ($A_3$ or
$A_4$) singularities. In this work we verify the existence of the two $A_4$
points by numerical simulations, observing subdiffusive behaviour of the
mean-square displacement and logarithmic decay of the density correlators.
Surprisingly, we also discover a novel dynamic behaviour generated by the
competition between the two higher-order singularities. This results in the
presence of special loci along which the dynamics is identical \textit{at all}
length and time scales.
