I know nothing about this incident other than as presented here but thought I might stick a couple of pence worth of technical info into the pot. "Modern" fire detection systems are much less likely to give false warnings tha older sytems. There are two independant fire detectors (as well as overheat detectors) and in normal operation both loops must detect a fire or overheat to present the indications to the crew. It is possible for a single loop to develop a fault which will not present a fire warning but will cause the system logic to automatically re-arrange itself so that the remaining good loop now operates as a single loop system and this can then present fire warnings if one subsequently occurs. (most MELs allow for dispatch in this config). Additionally, "most" Fire warnings are caused by hot gas leaks and do not necessarily involve flames at all BUT if it is hot gas then the warning will nearly always extinguish as thrust is reduced which of course will cause the temp of the gas leaking to reduce and will certainly disappear when the engine is shutdown. If fire warning indications persist after the appropriate drills then it is either a real fire or a false warning (subject to above comments). The latter is of course possible when the system is operating as single loop as mentioned.
Don't know if this helps or confuses.