Mostly due to her dislike of getting wet (and the smell of soap, which is easily remedied by very plain non-perfumed soap as most commonly preferred by jägers who largerly and bregrudgingly (with variation of opinion) accept that bathing is occasionally a good idea) Lilya generally dislikes bathing. Her hair gets wet and then says that way for several hours due to it's length, she's wet and cold and damp, and the whole process feels mundane and like a waste of time.
Unfortunately some pack-mates get to the point of deciding either the grime, smell, hair grease or multiple of the above are too much. Thus is she forced into taking a bath. With citrus soap if she doesn't cooperate (floral would just be too cruel.) Her hat (either kept hostage or kept safe) is safely stowed away from the water. Her hat and water don't mix well. In this case, that place is Barsin’s belt. She knows he won't take it cause it doesn't even fit over his horns well. And, frankly, its a rather plain hat. (But more importantly its her hat.)
And I just like Gkika. In the original planning of Lilya's backstory/first few years as a jäger, pre-general Gkika had a significant role cause it's my self-insert and projecrion, I can do what I want. Sakd backstory has since changed and evolved, but honestly, Gkika had to have gotten the nickname/title Mamma from something/somewhere. I like the idea that by jäger standards, she's somewhat the mother-hen type. Which involves more metaphorical pecking ie knocking of heads, but still incorperates 'being reasonable and responsible' (comparatively) and care (violent though it may be). Even though her time line and story might've changed a bit, Lilya does still get kinda attached to Gkika in some way or another.
Barsin, the green one of course, is from the same generation/group that took the draught as Lilya, so they've known eachother a decently long time and get along well. His name is short for Barsinufious.