Rsync : Preserve date after copying

I have an issue with files attributes since I changed my company's file system from an old ubuntu 12 to a Synology NAS.

I initially mounted NFS shares on my old file system, then with cp I copied everything, everything was ok But i forgot to preserve the attributes, and we lost especially the date, the dates became the copy date, and not the real date.

I found the rsync command rsync -vrt --size-only /src /dest But some files have been modified since the copy, and the command overwrite them...

For example i tried to create a file on the old file system, copying it on the new synology then modifiyng the file on synology. When i use rsync, the file is erased and replaced by an old version.

I am looking for a command, to copy timestamps for files only when the file have exactly the same size (haven't been modified)

No mater if it takes long time

Edit : I tried the proposed solution with a test directory, no error but no any changes:

old_base=/home/samba/shares/projects/test      
new_base=/mnt/NAS  
cp $old_base $new_base -rv
        `/home/samba/shares/projects/test/1201.txt' -> `/mnt/NAS/test/1201.txt'
        `/home/samba/shares/projects/test/test.txt' -> `/mnt/NAS/test/test.txt'
        `/home/samba/shares/projects/test/1200.txt' -> `/mnt/NAS/test/1200.txt'
        `/home/samba/shares/projects/test/1202/1202.txt' -> `/mnt/NAS/test/1202/1202.txt'

Now my new base is the same with the old, but no timestamp like the reality I modify one file, adding content to simulate a file edited since i copied everything Then i apply your solution :

    root@xx:/mnt/NAS/test# cd $old_base; find . -type f | xargs ls -s | sort -k 2,2 >/tmp/old
    root@xx:/home/samba/shares/projects/test# cd $new_base; find . -type f | xargs ls -s | sort -k 2,2 >/tmp/new
    root@xx:/mnt/NAS# comm -12 /tmp/old /tmp/new | while read size filename; do touch ${new_base}/${filename} -r ${old_base}/${filename}; done
    comm: file 2 is not in sorted order
    comm: file 1 is not in sorted order

I have no error, but no any changes (i am not sure what do you mean by ${filename} but i also tried to replace it by ${new_base}/*

Edit : 22/07

I have no any result when i use

comm -12 --nocheck-order /tmp/old /tmp/new

Outline of how I'd proceed: List the two hierarchies into a file with size. Sort, and compare the size files with "comm -12" to give a list of file names that have the exact same size. You can then either generate a list of timestamps and compare, or just feed that list into a "touch -r" command to duplicate the timestamps. Please don't blindly cut'n'paste these commands -- make sure you understand what they're trying to do.

Unless you've got a huge number of files, this shouldn't take that long.

Some simple (untested) commands that should help:

old_base = base directory of hierarchy with old files, desired timestamps

new_base = base directory of hierarchy with new files, improper timestamps

cd $old_base; find . -type f | xargs ls -s | while read size filename; do echo "${filename} ${size}"; done | sort >/tmp/old_list

cd $new_base; find . -type f | xargs ls -s | while read size filename; do echo "${filename} ${size}"; done | sort >/tmp/new_list

Try just doing comm -12 /tmp/old_list /tmp/new_list and looking at the output. If it's not filenames that need timestamps changed, we need to figure out why not.

comm -12 /tmp/old_list /tmp/new_list | while read size filename; do touch ${new_base}/${filename} -r ${old_base}/${filename}; done