BMW E39 EWS-DDE Synchronization: 3L Engine, Bosch 5 Plug Driver & Weak Battery Impact

3 liter engine, Bosch 5 plug driver.
Short question - can the synchro between EWS and DDE in this old stain due to a weak battery?

Thanks.
They attracted such a stain that it went out while driving, at the moment of starting a yellow arrow pointing to the left lights up on the clocks. "Immobilizer", "Unable to start engine" and "fuel pressure too low for starting" errors in DDE. Zero errors in EWS.
No immo will turn off a running engine, I know that.
In this, a cylindrical fuel pump fell. Dry in the filter.
Pump replacement, synchro DDE with EWS and smoke stain; but I don't like things that are unknown.
My theory is this - went out - they turned to the stop - the battery went down - the synchro went.

I was doing this on MoDicIII
Thanks, buddy for your help.