On our recent trip, the EB was quite late getting into CHI. Sleeper pax were given a choice of three items from the regular menu (not including the steak). The purpose of the reduced menu was to get all sleeper pax served *quickly* because they needed to serve all the coach pax their stew after us. I saw no evidence food was running low, but they may have run out of one of the three choices before all frist-class pax were served.
On the SL, we weren't late, but I chatted a bit with one of the OBS, who said that all the food they would or could get for the trip was already on-board (including the AmStew). If anything ran low or out, they would just serve what they had. I will say the cafe was almost totally out of food at the end of the trip! I know the SL is different from the EB (maybe not for much longer...), but it seems a good idea to have a stash of energy bars, or dried fruit and nuts, somewhere in your luggage for emergencies.