  We consider critical percolation on a supercritical Galton-Watson tree. We
show that, when the offspring distribution is in the domain of attraction of an
$\alpha$-stable law for some $\alpha \in (1,2)$, or has finite variance,
several annealed properties also hold in a quenched setting. In particular, the
following properties hold for the critical root cluster on almost every
realisation of the tree: (1) the rescaled survival probabilities converge; (2)
the Yaglom limit or its stable analogue hold - in particular, conditioned on
survival, the number of vertices at generation $n$ that are connected to the
root cluster rescale to a certain (explicit) random variable; (3) conditioned
on initial survival, the sequence of generation sizes in the root cluster
rescales to a continuous-state branching process. This strengthens some earlier
results of Michelen (2019) who proved (1) and (2) in the case where the initial
tree has an offspring distribution with all moments finite.
