I’m posting here my experience.
I recently installed on one computer 3 distributions : Debian, Fedora, Manjaro.
At my first try I installed Debian as last distribution, meaning the operating GRUB was the Debian one, so I had issue on Initramfs process. More Precisely, Debian’s GRUB “LINITRD” variable took intel-ucode.img as image to put in the file 30_os-prober to generate the initrd.
The only way to fix this in this configuration is to change grub.cfg manual and replace intel-ucode.img with the right initrd image (depends your config and you can copy the one for the fallback) each time grub is changed, or to boot in fallback mode.
Finally the most convinient way to make them live, is to install them Manjaro at last, beacause the main hardware GRUB which loads all distros boot is the last you installed. In that way you can update each distros as you want and everything works fine.
Wish it can help.