Do you realize that whenever you fob is in "contact" with your bike it is draining the battery. Your battery will last much longer if you store it out of range of your bike.
Actually the bike is still "actively" seeking a signal with the fob regardless if that fob is within range. That is why keyless ignition vehicles of all types will usually drain their batteries faster than antique keyed vehicles.
Although it is possible that the fob and bike systems are
more active and drain the batteries very slightly more when they do "see" each other.
The fob is also always active even when it is out of range or, in my case, encased within a RFID pouch. I needed to replace the battery in both my fobs about 3 months ago. Both of them went dead at the same time even though my spare fob spends 99.9999% of the time within a RFID pouch sitting in the storage box since I first brought the bike home.
If both ends of the keyless system, the bike side and the fob side, were not actively sending or seeking a signal from one to the other then how would the bike know that the fob is there and respond. It is a constant ON system. One thing I do to prevent a little of the battery drain is to turn off the Noodoe Welcome light option (Noodoe phone app vehicle settings). That way I don't have the Noodoe screen turning on every time I walk near the AK if I do have the fob in my pocket and just walking in the garage or around the campsite when I am traveling. Screen lighting uses a lot of power compared to other "radio" signal devices.
BTW- even modern cars/SUVs that still use keys have other systems that are constantly ON and run down the battery if not driven/ridden at least every 1-2 weeks. I remember having a car a long time ago that I could let sit and not drive for 2-3 months and the battery was still strong to start it. Now my newest car, 2017 Legacy, which uses an old style key ignition will end up with a dead battery if I let it sit for 2 weeks and is too weak to start the engine if it sits for about ONE week. So I keep all vehicles on a Battery Tender if not used every other day at least. When I am on a 5 day mc camping trip with my AK I find that I will go for a ride in the area at least every other day so the battery is not yet run down enough to need the BT connected while at the campground... but I bring my BT Jr. anyway just in case. It's small and sits under the seat.