What is the small blue wire in my 240V air conditioner cord?

I’m looking to replace the plug on a window air conditioner unit that uses a 6-20 240v plug. However, in addition to the cord having two hots and ground, there’s a small blue cable that I’m not sure what it’s hooked up for. Is it for the circuit breaker (leakage current sensor, or LCDI; it’s similar to a GFCI receptacle with the test & reset buttons but apparently for a different use) built into the plug, and if so can I bypass it? And if not, what is it for, and how do I account for it on the new plug, which just takes hot, neutral, and ground? I realize the risks of just bypassing it, but I’m desperate.

The air conditioner probably around 15 years old, give or take. At least 13 years. It’s a Whirlpool, the model is ACQ244XR2.

The longer story, I’m hoping this will fix the air conditioner. It simply does not turn on anymore; there’s no display, and the controls don’t work (it’s all digital). The test/reset breaker on the plug actually still seems to work okay, but of course, there’s no power. I’ve tested the plug with a multimeter and everything tests okay (there’s the occasional jump to 1000 when I wiggle the lead connected to a hot but I’m assuming that’s user error or from it being a cheap multimeter). All in all, I can’t afford to replace it or hire an electrician so I’m hoping this is the only issue. Any help at all would be greatly appreciated.