  Bootstrap percolation is a cellular automaton modelling the spread of an
`infection' on a graph. In this note, we prove a family of lower bounds on the
critical probability for $r$-neighbour bootstrap percolation on Galton--Watson
trees in terms of moments of the offspring distributions. With this result we
confirm a conjecture of Bollob\'as, Gunderson, Holmgren, Janson and Przykucki.
We also show that these bounds are best possible up to positive constants not
depending on the offspring distribution.
