There are two fare buckets in First: 'A', which is Flex and 'F', which is Flex+. Flex+ is substantially more expensive than Flex (although Flex is by no means 'cheap'!).
What you should do first is check if the available seats are Flex or Flex+. This you can do via a dummy booking (do a dummy booking for a return trip as that might influence things). If there are Flex seats available, there is no real chance the fare will drop as there is no lower fare in existence. If there are only Flex+ seats available, there is a chance that they will release Flex closer to the time, depending on what the algorithm they use thinks about the probability that they can sell them as 'A' or 'F'. To be honest, in my experience chances are not all that good, but you never know. If you get a subscription on Expert Flyer, you can set an alert that will let you know if / when the 'cheaper' seats are available. Expert Flyer costs $9.99 per month and you can 'switch it off' easily when you don't need it (and switch it on again when you do). Although it's functionality has reduced in the past few years (on a number of airlines, including EK you can't search for awards anymore), I still think it's good value in the months that you need the information. They also have a 5 day free trial period (and no, I am in no way connected to EF).