If you were to navigate to Ghosts Of The Deep and look at how many boosts you have towards the exotic weapon dropping, it doesn't show a number towards your boosts. I can't tell if the boosts themselves actually apply to GOTD(I don't have the exotic btw,) so I can't tell any more info about that. Additionally, if players are trying to get the final memory for the triumph towards an exotic drop boost after completion of the final boss, there is a very small window due to the cut down mission end timer of just 1 minute. That means players won't be able to get the memory if they aren't fast enough. I'd recommend 3 minutes to get the memory.