I have tried to reproduce your work and have written the rules as such. I hope they are free of careless error
WolframModel[{
{{1,2},{2,1},{2,3},{3,2},{3,4},{4,3}}->
{{1,2},{2,1},{2,3},{3,2},{3,4},{4,3},{22,2},{2,22},{22,23},{23,22},{23,3},{3,23}},
{{1,2},{2,4},{4,2},{2,3},{3,2},{4,6},{6,4},{5,6},{6,5},{6,7},{7,6},{6,8},{8,6},{7,9},{9,7}}->
{{1,2},{2,4},{4,2},{2,3},{3,2},{4,6},{6,4},{5,6},{6,5},{6,7},{7,6},{6,8},{8,6},{7,9},{9,7},{4,111},{111,4},{111,7},{7,111}},
{{1,2},{2,4},{4,2},{2,3},{3,2},{4,6},{6,4},{5,6},{6,5},{6,7},{7,6},{6,8},{8,6},{7,9},{9,7},{7,10},{10,7},{11,10}}->
{{1,2},{2,4},{4,2},{2,3},{3,2},{4,6},{6,4},{5,6},{6,5},{6,7},{7,6},{6,8},{8,6},{7,9},{9,7},{7,10},{10,7},{11,10},{4,111},{111,4},{111,7},{7,111}}
}
,{{1,2},{2,1},{2,3},{3,2},{3,4},{4,3},{4,1},{1,4}}
, 5, "FinalStatePlot"]
Those result in this rule plot:

It is distorted by the graph plotting style but the topology looks correct.
This rule with these initial conditions:

follows this cancerous evolution.




The problem it that r1 can make it grow in any direction and can start to grow from any square. I already devised an approach based on hyper graphs to trim the space of allowed extensions in something close to a plane. Do you want to revise your strategy?
Do you have a recent version of Mathematica available?