Create an audio cable with 100k ohm resistors
From their FAQ:

Can't hear music when using Mighty with an auxiliary cord / Does Mighty support AUX cord?
Mighty doesn't currently work with every type of auxiliary cord (which certain Bose headphones act as). The root cause is somewhat technical - it has to do with Mighty's headphone jack and the requirement to detect the aux cord plugin electronically. The good news is that this can be resolved through a software update and our tech team is currently working on this feature. We plan to roll out in the next month. [No date for the post, of course. It's a rolling month or &quot;real soon now&quot;.]

You might be able to fool it by wiring a couple of hundred ohms between each signal wire and ground. I couldn't find any reference to 300 Ω you mentioned.