I may be getting too far down in the weeds on this, but... for the sake of discussion:
Based on what I can see, since it is not possible to hold the artifact in person myself, I am thinking maybe the major stain on the bill and back of the cap might not entirely be sweat? It may be a bit of sweat, mixed with a petroleum product, aviation fuel perhaps. For example, on some aircraft there is/was a walk-around inspection step which involves bleeding off possible water in fuel lines/tanks through a petcock underneath wings... more than once the wind or me placing my face too close to that procedure ended up getting stuff where it wasn't ideally supposed to go.
The inner sweat band should, it seems to me, evidence a LOT more sweat stain for sweat to have migrated so thoroughly out onto the bill, from what I can see there. That sweat band just doesn't seem to show evidence of very much actual sweat?
The black, or dark, stain on the back of the cap looks like, almost exactly, the kind of crud I got on my own mechanic's cap just from having greasy hands and pulling back on it to get it away from my eyes during maintenance.
Maybe not a big huge deal, of course, but there is a world of solvent difference between removing sweat and removing petroleum crud.
Also, what Gil said about the probability of the bill being made of cardboard seems very high, so it could easily fall apart or come to pieces with enough moisture involved. If the bill can be bent or curled and holds its shape, then it is probably cardboard. If it is made of some sort of plastic or phenolic, then it will resist being curled more.
Edited by Bluehawk, 17 May 2018 - 04:50 AM.