AIX Korn Shell - compare date in yyyyMM format

I'm searching for compare two date in AIX. This is the first date in yyyyMM format:

three_months_ago=$(( $(date +%Y%m) - 3))

And I get the second date from filesystem with concatenation of folder name:

for dir in /p22_satos/satos/data/*                                                     
do                                                                                 
 echo ${dir}  
 year=$(basename ${dir})
 if [[ $dir = "/p22_satos/satos/data/static" ]];
then
    print "Static dir should not be deleted"
else
    for   subdir in ${dir}/*
     do
       echo ${subdir}
       month=$(basename ${subdir})
       if [ $month -ge 10 ];
       then
         echo ${year}${month}
         echo $three_months_ago
         if [ ${year}${month} -ge $three_months_ago ];
         then            
           echo "The directory should not be deleted"
         else
            echo "The directory should be deleted"
         fi
       else 

        echo ${year}"0"${month}
        echo $three_months_ago
        if [ ${year}"0"${month} -ge $three_months_ago ];
         then            
           echo "The directory should not be deleted"
         else
            echo "The directory should be deleted"
         fi
       fi

       #rm -f $subdir
     done
fi

The value for $three_months_ago is 202103. I've folder 2020 with subfolder 12, so the cycle make 202012 date dynamically.
Why ${year}${month} and ${year}"0"${month} are always greater than $three_months_ago and the if [ ${year}${month} -ge $three_months_ago ] fails?