Toom's north-east-self voting cellular automaton rule R is known to suppress small minorities. A variant, R +, is also known to turn an arbitrary initial configuration into a homogeneous one (without changing the ones that were homogeneous to start with). Here it is shown that R + always increases a certain property of sets called thickness. This result is intended as a step toward a proof of the fast convergence toward consensus under R*. The latter is observable experimentally, even in the presence of some noise.