Hoople
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One would hope that the ground for the radio is not relying on that coax cable connection only.
I agree 100%.
The fact that you can unplug the coax and the radio dies only supports the theory (missing ground in harness) even more.
Years and years ago radios, motors, etc would use the shell or housing for a Battery return path. It is Rare you see that these days. It would be insane not to have a ground wire in the Harness.
Hoop!