I don't know, but I guess it matters just as it does with other power cords. Is it worth the sound improvement? I guess only you can answer that.
FWIW, my highly optimized PC with linear PS has a generic power cable.
I also think it depends how the rest of the PC is powered. For example, if you provide ultra clean power to the JCAT, but the rest of the computer is supplied from a noisy PS, I suspect the power cord to the PS will make less of a difference than if the mobo and (assuming) SSD for the OS are each fed from a clean source.
But this is my guess. I suggest you post this question at the JPlay forum, where some folks have been experimenting big time with many combinations of power supply methods. I've seen you around computeraudiophile.com, but I don't think people there are as open minded about power supplies as on the other forum mentioned.
Let us know how it goes!