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In[]:=

Module[{k=3,r=1,cut=200,deep=5000,fun=Length,init,evo,ru,data,lt,test,mask,canonical,res},res=ParallelTable[SeedRandom[seed];init=FromDigits[Lookup[Association[Thread[Append[Permutations[CenterArray[1,2r+1]],CenterArray[0,2r+1]]->1]],RuleTuples[k,r],0],2];init={0,{0,init},1,fun[{{1}}]};evo=NestList[Function[item,ru=First[RandomRuleMutation[{First[item],k,r}]];data=CellularAutomaton[{ru,k,r},{{1},0},{cut,All}];lt=If[SameQ[#,0],True,cut-#+1]&[LengthWhile[Reverse[data],Total[#]==0&]];If[TrueQ[lt],item,test=fun[MatrixCrop[data]];If[test<Last[item],item,data=Union[Catenate[Partition[#,2r+1,1]&/@data]];mask=Lookup[Association[Thread[data->1]],RuleTuples[k,r],0];canonical=IntegerDigits[ru,k,k^(2r+1)];canonical={FromDigits[canonical*mask,k],FromDigits[mask,2]};{ru,canonical,lt,test}]]],init,deep];evo=Map[Rest@*First,SplitBy[evo,#[[2]]&]];evo=Map[DeleteDuplicates,SplitBy[evo,Last]],{seed,5000+{53,16,73,54,60}}]]

Out[]=

{{{{{0,523},1,1}},{{{39366,262735},2,2}},{{{39879,295535},3,3}},{{{28750815,17073007},4,4}},{{{5104666769346,86279151},6,6}},{{{5251112777070,98866159},9,9},{{5157232659513,90625023},9,9}},{{{5159428101333,99547135},14,14}},{{{6004478576529,132636671},26,26}},{{{6004478576535,133685247},50,50}},{{{6006804516645,134217727},92,92}}},{{{{0,523},1,1}},{{{39366,262735},2,2},{{39420,17040143},2,2},{{39426,84149007},2,2},{{60,17040143},2,2},{{6,262735},2,2}},{{{6973608174,1311311},3,3}},{{{188286399240,8684271},4,4},{{188315097054,25592559},4,4}},{{{195546946182,26641135},5,5}},{{{551208521022,100582127},10,10}},{{{2176050058530,129942527},12,12},{{1391523561744,134136575},12,12}},{{{1297425260157,134217471},21,21}},{{{1304398828959,134217471},22,22}},{{{1304399360400,134217471},23,23}},{{{2151687976404,134217727},97,97}}},{{{{0,523},1,1}},{{{54,262735},2,2}},{{{564859112382,17040207},3,3},{{564859112388,84149071},3,3}},{{{5648590769046,84149071},4,4}},{{{3113727207333,85230447},5,5},{{101145484842,9732847},5,5}},{{{5742762646146,92570607},6,6}},{{{5742749359797,92574687},7,7}},{{{690398940300,30841855},9,9},{{5805511656081,96902111},9,9}},{{{6726030532899,133624831},10,10}},{{{6757411592508,132445183},11,11}},{{{6757623639573,132633583},35,35}},{{{6570501315018,134217727},46,46}},{{{6570501137871,134217727},79,79}}},{{{{0,523},1,1}},{{{387421281,275039},3,3},{{387421272,266847},3,3},{{792,266847},3,3}},{{{14353101,2396927},5,5},{{430471404,2429695},5,5}},{{{241032943980,15061759},6,6},{{220112294679,12965599},6,6}},{{{251493354288,12965631},7,7}},{{{2228457065121,65443839},10,10}},{{{1946027528640,65443839},12,12},{{1956530928564,65509375},12,12}},{{{1954469823219,134209535},25,25}},{{{1950983038818,134209535},29,29}},{{{1952145300285,134209535},42,42}},{{{1952140517316,134217727},43,43}},{{{1948653555768,133144575},60,60}},{{{1955585672172,134217727},140,140}}},{{{{0,523},1,1}},{{{54,262735},2,2},{{39420,17040143},2,2}},{{{564859112382,17040207},3,3}},{{{5648590769046,84149071},4,4}},{{{3106724941203,84181871},5,5}},{{{3954027899553,119835503},7,7}},{{{3995874095496,124050303},9,9}},{{{4006592729025,124181375},11,11},{{3982180455249,121553791},11,11}},{{{4013566474974,125754239},13,13}},{{{4201853895510,133684223},20,20}},{{{4201857261303,133692415},31,31}},{{{4201857263490,134216703},34,34}},{{{4201857263409,134216703},44,44}},{{{4201856731968,134217727},85,85}},{{{4201856850066,134217727},107,107}},{{{4201856791017,134217727},144,144}}}}

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Last/@%

Out[]=

{{{{6006804516645,134217727},92,92}},{{{2151687976404,134217727},97,97}},{{{6570501137871,134217727},79,79}},{{{1955585672172,134217727},140,140}},{{{4201856791017,134217727},144,144}}}

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