Percolation of even sites for enhanced random sequential adsorption

Consider random sequential adsorption on a chequerboard lattice with arrivals at rate 1 on light squares and at rate λ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites black, and the remaining squares white. Independently at each meeting-point of four squares, allow diagonal connections between black squares with probability p; otherwise allow diagonal connections between white squares. We show that there is a critical surface of pairs (λ,p), containing the pair (1, 0.5), such that for (λ,p) lying above (respectively, below) the critical surface the black (resp. white) phase percolates, and on the critical surface neither phase percolates.