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 $\lambda$ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites {\em black}, and the remaining squares {\em 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 $(\lambda ,p)$, containing the pair $(1,0.5)$, such that for $(\lambda ,p)$ lying above (respectively, below) the critical surface the black (resp. white) phase percolates, and on the critical surface neither phase percolates.