mcomer

The fob can be put in Battery Saver mode by holding the Lock button down and then pressing
the fob Unlock button twice. Watch the small red guide light flash four (4) times. The fob is now
temporarily turned off. Going for a jog or beach activities? Don't want to take the heavy, expensive,
non-waterproof fob with you? Stand at the open driver's door, perform the fob cadence, pull the
mechanical key out of the fob and conceal the main fob under a floor mat. Lock it manually and go.
When you return use the mechanical key to gain access, recover the fob and press any button to
wake it up. This feature started with the current style fob, all four button in a row. The previous style
that came out in 2010 with some buttons offset does not have the power down function.

dibl

Regarding the OP's question, I take it that the car is in "listening" or "fob detecting" mode 24/7, to detect the presence of a fob. There are two identical fobs and potentially two drivers. Once the currently detected fob moves out of range of the car's detection capability, the car can't know whether the fob is on or off, nor when the next fob might come into range. Therefore, by logic, the car's fob detector cannot ever be turned off, and thus zero reduction in the parasitic drain on the car's battery. Or so it seems to me.

mcomer

The car's Smart Access can be turned off, traditionally for drivers with heart pacemakers that
are not to be subject to the radiation of smart keys and microwave ovens. Such a driver would
then need to manually unlock the car and hold the fobthisclose to the start/stop button to work.
Menu item on newer Lexi, Techtool Dealer service on older models.

Cocal

I have a pacemaker and not the one good for MRI, but have no problem with the FOB. Pacemakers are affected by magnetic frequency and not microwaves.
If one has the latest one they aren't affected at all. I wanted to have an induction stove, but because of my pacemaker the Dr said, better play safe and not have it.