In[]:=
Select[ParallelMapMonitored[WolframModelTest[#,{{1,1},{1,1}}]&,Table[{{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}},RandomWolframModel[{{2,2}}{{1,2}}]},1000]],(Max[#Sizes]>6&&Count[Sign[Differences[#Sizes]],-1]>3&&ConnectedHypergraphQ[#FinalState])&];
In[]:=
MakePictures[%]
Out[]=
{},
Clear
Copy

In[]:=
aa=Tuples[{EnumerateWolframModelRules[{{1,2}}{{2,2}}],EnumerateWolframModelRules[{{2,2}}{{1,2}}]}];
In[]:=
Length[%]
Out[]=
4672
In[]:=
Select[ParallelMapMonitored[WolframModelTest[#,{{1,1}}]&,aa],(AnyTrue[Differences[#Sizes],#<0&]&&ConnectedHypergraphQ[#FinalState])&];
In[]:=
MakePictures2[First/@GatherBy[%,#FinalState&]]
Out[]=
In[]:=
Select[ParallelMapMonitored[WolframModelTest[#,{{1,2},{2,3}}]&,aa],(AnyTrue[Differences[#Sizes],#<0&]&&ConnectedHypergraphQ[#FinalState])&];
In[]:=
MakePictures2[First/@GatherBy[%,#FinalState&]]
Out[]=

Pairs of 2,3 rules

More

Rule Index Testing