This one was in fact the rubber pip that shuts the lantern off. See how its all swollen with fuel. When new it would just barely protrude from the cup, now it wont allow fuel flow so the lantern won't light. When I tried to pull the pip out of the cup it tore apart, the rubber was just saturated with kerosene. The irritating thing is that the eBay seller I got the pips from denies theres anything wrong with them and now won't even reply to my messages.
Since the seller refuses to make things right I have no problem calling him/her out, so for the record don't buy a Tilley service kit from hootflipflopandfly on eBay, the pips will swell up in the presence of kerosene which is the proper Tilley fuel. The pips are poorly shaped and don't fit particularly well in the cups anyway. They were made with a cheap cutter and show significant hour glassing which leads me to believe they were homemade even though the seller claims they are "Genuine Tilley". Instead go to Harbor Freight and buy a punch set, then order a sheet of proper Buna-N rubber from McMaster Carr, you'll have enough material to make a lifetime of Tilley pips. I've had my sheet of rubber since 2013 and have used only a small corner of it.