Thanks, everyone!
this was working perfectly.

so to conclude, we didn't solve the variant delay issue, as it seems it's not really possible to solve it.
so the conclusion is that IDM isn't really doing a good job in capturing this type of stream.
FFMPEG did the job, and all files are synced.

I have 1 file that still has an issue, for some reason, FFMPEG returns an error saying:-manifest too large.
Does it mean that the MPD file is too long?
I didn't find any way around it...

Thanks a lot for all your help !