co.combinatorics - Path connected coloured sets on the squared paper

I believe that such a 3x3 square does not necessarily exist.

A counterexample would take the form of an infinite still life pattern in the life-like cellular automaton rule B123678/S34 (these rules are chosen so that the only patterns that remain stable are the ones in which the number of live cells in each 3x3 box is 4 or 5). Additionally, both the live and dead cells of the pattern should be connected.

But as the following partial double spiral shows (copy and paste it into Golly to view and test) it's possible to form partial double-spiral patterns that, at least in the center of the pattern, have the desired properties. I don't see any good reason why it shouldn't be possible to continue the spiral infinitely.

x = 31, y = 31, rule = B123678/S34
14b4o$12b3o2b3o$10b3o6b3o$8b3o3b4o3b3o$6b3o3b3o2b3o3b3o$5b2o3b3o6b3o3b
2o$5bo2b3o3b4o3b3o2bo$4b2ob2o3b3o2b3o3b2ob2o$4bo2bo2b3o6b3o2bo2bo$3b2o
b2ob2o3b4o3b2ob2ob2o$3bo2bo2bo2b3o2b3o2bo2bo2bo$2b2ob2ob2ob2o6b2ob2ob
2ob2o$2bo2bo2b2obo2b4o2bo2bo2bo2bo$2bo2bo2bo2bob2o2b2ob2ob2ob2ob2o$b2o
b2ob2ob2o2bo2bo2bo2bo2bo2bo$b2ob2ob2ob2ob2ob2ob2ob2ob2ob2o$bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo$2ob2ob2ob2ob2ob2o2bo2bo2bo2bo$o2bo2bo2bo2b2o2bob2ob
2ob2ob2o$2ob2ob2ob2o4b2obo2bo2bo2bo$bo2bo2bo2b6o2bo2bo2bo2bo$b2ob2ob2o
3b2o3b2ob2ob2ob2o$2bo2bo2b3o4b3o2bo2bo2bo$2b2ob2o3b6o3b2ob2ob2o$3bo2b
3o3b2o3b3o2bo2bo$3b2o3b3o4b3o3b2ob2o$4b3o3b6o3b3o2bo$6b3o3b2o3b3o3b2o$
8b3o4b3o3b3o$10b6o3b3o$12b2o3b3o!

Here's a screenshot: