KRK 10s wants to be touched

If by "center of its backplate", you mean the plate with the input jacks, the pressure applied may be flexing the circuit board enough to cause a bad solder connection to make connection.
After removing the jack pate, with an ohm meter, you can check between connectors and circuit board traces downstream to determine where the problem is.
Re-flowing the bad solder joints could fix the problem.
 
As mentioned above, likely cold solder or connection.

Best bet is to take it apart, turn it on and prod various parts (with an insulated stick of sorts) until you identify the connection / component!

It may also be identified by flexing the PCB, which could fixing slightly harder - but not impossible 🙂