Yes, we solved it ! And here's how.

Indeed, the almighty oscilloscope proved that the thought from the previous log is correct. The D5-R14-R15 divider does not drop below 0.4V long enough to turn the NCP705 off.

And here is the work-around: a 1uF capacitor.

We also modified the value of R14 (was 47K, now is 100K) and the reference voltage divider. The power-down did not work reliably at low battery voltages.