There's a tradeoff with wired headphones. Sometimes they act like their own antenna, and if you live in an area with very poor reception, then that becomes a huge problem. In that scenario, bluetooth earbuds are actually preferable, until they run out of battery mid-call.
Not saying this is impossible, but it doesn't make sense to me.
The phone antenna isn't a straight piece of wire entwined with your cable (probably not even parallel with it), and your cable isn't connected to the antenna port.
Maybe they mean the noise you can hear from the phone as it communicates. I wonder if it is more pronounced when reception is bad as it searches for other towers? I had wired headphones + mic with an amplifier for a while and if I put my phone anywhere near it you'd hear the classic GSM buzz.
So they do share ground, which is where the interference comes from. You might find sometimes even stereo sound speakers pick up a ticking noise right when you receive a phone call. This is pretty common with speaker systems, wired headphones being a type of speaker system.
Also, a lot of SoCs with FM antennas use the 3.5mm ground as the antenna, so they are wired into the SoC in such a way that they can receive undesired interference.
