  Dual-unitary circuits are a class of locally-interacting quantum many-body
systems displaying unitary dynamics also when the roles of space and time are
exchanged. These systems have recently emerged as a remarkable framework where
certain features of many-body quantum chaos can be studied exactly. In
particular, they admit a class of ``solvable" initial states for which, in the
thermodynamic limit, one can access the full non-equilibrium dynamics. This
reveals a surprising property: when a dual-unitary circuit is prepared in a
solvable state the quantum entanglement between two complementary spatial
regions grows at the maximal speed allowed by the local structure of the
evolution. Here we investigate the fate of this property when the system is
prepared in a generic pair-product state. We show that in this case the
entanglement increment during a time step is sub-maximal for finite times,
however, it approaches the maximal value in the infinite-time limit. This
statement is proven rigorously for dual-unitary circuits generating high enough
entanglement, while it is argued to hold for the entire class.
