The two compartments specially designed for the batteries still have about 1 cm of 'air' in the width. Does that fit?
2 Wearing comfort is an individual thing. The backpack is relatively narrow, has an easily adjustable strap and is well padded on the back.
Tip: There are videos on youtube that demonstrate how to use the backpack.

Hello everyone, I received my Manfrotto bag today. The width of the battery fits perfectly in the new bag. The bag has already been improved.

I've also seen a video review of this bag and he tries to have the battery entering in the bag's pocket without success

Great, thank you very much. Unfortunately, there is very little information about this backpack on the internet. I have already seen the videos, thank you. But they are all from the manufacturer without any real appreciation.
It's a shame that Manfrotto can't clarify the compatibility with the P4. An expensive experiment if it doesn't fit.

Yes the P4 fits as it has the same dimensions