Edit by @moderators: 
This post contains instructions that can cause severe harm to you.
So welcome in the club of broken pogo pins 
I am lucky that I only have trouble with these pins at work.
There is one thing you could try before you swap your core module.
It is a dangerous action, but if you do not have anything to loose it is worth a try.
I guess you have (a) broken solder joint(s) on the pogo pin(s). Resoldering by hand would be hard, moreover most people do not have access to professional soldering equipment. But you do have an oven in your kitchen I guess.
So the melting point of solder tin is around 200°C, so you could try a self soldering session.
Dissemble anything you could from your core module. Heat up you oven to 220°C and put the core module with the pins up (!) in the oven for ~10 minutes. (! = You do not want the pins to get lost by the gravity if you put your module with the pins down, smaller parts, like resistors will be hold by the surface tension of the tin).
To check if the solder melted in the process time, you can use a pen on an empty solder joint before you put it in the oven. When it was melted the marking should be gone. If it is still there leave it for some more minutes in the oven.
Caution:
When the core module is hot and you take it out of the oven, beware shocks to the module. the solder is liquid and the parts are swimming on the liquid solder. If it will fell down it will be unusable due to the parts are shifted. Then you successfully made a Fairphone to a Shiftphone 
I know it is a little nerdy to “cook” your phone to function, but there are some circumstances where that helps.
You could try a google search for more feed.