  This article is concerned with a conjecture by one of the authors on the
formation of dispersive shocks in a class of Hamiltonian dispersive
regularizations of the quasilinear transport equation. The regularizations are
characterized by two arbitrary functions of one variable, where the condition
of integrability implies that one of these functions must not vanish. It is
shown numerically for a large class of equations that the local behaviour of
their solution near the point of gradient catastrophe for the transport
equation is described locally by a special solution of a Painlev\'e-type
equation. This local description holds also for solutions to equations where
blow up can occur in finite time. Furthermore, it is shown that a solution of
the dispersive equations away from the point of gradient catastrophe is
approximated by a solution of the transport equation with the same initial
data, modulo terms of order $\epsilon^2$ where $\epsilon^2$ is the small
dispersion parameter. Corrections up to order $\epsilon^4 $ are obtained and
tested numerically.
