Hard cold starting just suggests they are not working. The GP themselves can be in working order, but not getting the supplied voltage.
I'd recommend removing the wire that feeds the GP's and measure voltage when the GP dash light is present. If no voltage this wire can be traced back to find the offending connection, fuse, relay, module, etc... (I'm fairly certain the voltage at the GP can not be measured unless they are removed from the circuit.)
Don't "Look" at a connection and say its good. Verify by voltage drop test or disassemble, sand/wire brush to ensure metal to metal contact. Many forum members have found issues with fuses not making good contact or corrosion within the fuse holder. It's always worth trying to disassemble/reassemble wire connectors a few times if difficult to clean otherwise.
Do you have a wiring diagram and description of how the GP circuit functions (found in workshop manual)? I have one for the 855D and it is very helpful, but likely not the same as your HPX.
Good luck and report back what you find.
2012 JD855D OPS Black Poly Roof, Deluxe Cargo Box, Front and Rear Protection, Power Lift, Front Hood Rack, Rear Screen (homemade), 800 Watt Inverter, 2500lb Winch installed in cargo bed, Front and Rear LED auxiliary lighting (4" square 27 watt), Front and Rear CV Guards, Koplin Side Mirrors