I did notice that once the player wakes up from a downed state, sometimes in the bottom left will appear 'Put Away' with a UI symbol for Left Click.
Reference: https://i.gyazo.com/b7a177ebda7ae53a6fb842b02c70d663.png
From doing more testing, it seems that the Low Ready function stops working sporadically when playing / doing random things (after placing bipod, going prone, switching character etc)..