I have had a 45 minute phone call using the Fairphone modular earphones without any problems on my FP1 just this sunday and am listening to music 
using them on an FP2 (original 2MP top module) right now. On the FP2, volume control as well as pausing works on the default Music app (under the latest /e/ version).
Further testing
Fairphone Open OS 19.08.1.
Reduced functionality. Plugging in sometimes gets recognized, sometimes not. When the connection is there, both audio playback works and the microphone picks up my voice (in Sound Recorder), but remote control 
input is without any effect (neither pause/skip nor volume) in the default Music app although the notification bar shows a headset symbol.
Maybe Android 7 vs. Android 9 makes a difference here.
Sailfish OS 3.2.1.20 (Android 6?)
Very limited testing possible for me only (no music app pre-installed, resorted to just testing optional system sounds). Audio playback works, volume control works, pause does not work.
Ubuntu 16.04
Same result as Sailfish OS: Audio playback works, volume control works, pause does not work. Hit and miss when it comes to plug recognition.
LineageOS 17.1
Audio playback works, volume control works, pause sometimes worked … in a very special way (vague theory: once you have paused a track, the same button will only work again to resume play after you’ve waited like 10 seconds – otherwise it won’t respond at all). Functionality of the remote control is generally hard to predict, it will sometimes even skip forward or back a track when using the volume buttons. Microphone works. Hit and miss when it comes to plug recognition.
Microphone tests all done with both the primary and the secondary microphone simultaneously pressed shut with my thumb/finger – to make sure it's the headset phone that picks up my voice, not the FP2's built-in ones.