Yes that's likely the issue - sometimes an easy way to test that wire is to activate the controller manually and see if the brake lights on the tow vehicle light up. If they do, then the brake signal wire is hooked up properly - if not, it's not.
I can do some diagnostics right through the P3 and it's showing 0 volts on the brake light signal when my foot is on the brake..
Interesting, that must be something they added to the P3 software after I got mine as I'm pretty sure the ability to see the signal voltage isn't part of mine. It will show output voltage to the trailer brakes of course, but not incoming signal voltage. That's a handy diagnostic tool, for sure.
Menu > Help > Troubleshoot
Then you can scroll through different signals. Battery voltage, brake output voltage and amperage, and brake light signal voltage.
Do you have that feature??