Many chips don't go to sleep mode until the VDD has dropped quite a bit.
See:
The 2V63 value might be the ideal value on their VDD/IDD graph, with most headroom for battery voltage droop, but lower voltages (e.g. 2V4) might work just as well.
Textbooks show zero current up to Vf but I suspect Vf drop depends on current, even before then.
It should be a quick experiment to see what happens.