  We consider the soft-core Widom-Rowlinson model for particles with spins and
holes, on a Cayley tree of order $d$ (which has $d + 1$ nearest neighbours),
depending on repulsion strength $\beta$ between particles of different signs
and on an activity parameter $\lambda$ for particles. We analyse Gibbsian
properties of the time-evolved intermediate Gibbs measure of the static model,
under a spin-flip time evolution, in a regime of large repulsion strength
$\beta$. We first show that there is a dynamical transition, in which the
measure becomes non-Gibbsian at large times, independently of the particle
activity, for any $d \geq 2$. In our second and main result, we also show that
for large $\beta$ and at large times, the measure of the set of bad
configurations (discontinuity points) changes from zero to one as the particle
activity $\lambda$ increases, assuming that $d \geq 4$. Our proof relies on a
general zero-one law for bad configurations on the tree, and the introduction
of a set of uniformly bad configurations given in terms of subtree percolation,
which we show to become typical at high particle activity.
