What about a rug in between to hide the cable??
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What @erik_squires said.
Or take the long way around: either around the back of the listening position or a wire route that exits the room next to the existing power outlets, around the back of the hearth and re-enters next to the other power outlet on the other side. Assuming that is outside the house, a conduit (flexible) could be used.
I would be studying the base of the hearth and any molding that is used - or could be used - as a cover for the wiring. Obviously, if the bottom of the hearth is at floor level this wouldn't work.
Then again, I would also use any excuse that seemed reasonable to buy two Bluesound Powernodes. ;)
In any event, without pictures of the installation, I can only offer guesses.
Hope this helps. Good luck and congrats on the new digs.
Here's another option. Use a music server than runs LMS (Logitech Media Server). Then, get two Raspberry Pi players and set them up with the PiCorePlayer OS and a suitable DAC HAT on each one, and enable the WiFi networks. Then, in the LMS software, sync the two players. On the RPi player on the left side, run a RCA cable only from the left output into a monoblock amp. On the right side, use only the right output into that mono amp. This would allow you to stream your local library of ripped files as well as use Qobuz, Tidal or another streaming service.
Of course, you'll want identical RPi boards and HATs (don't have a RPi 3 on one side and a 4 on the other, etc.) along with the same software in order to eliminate any time differences between the two channels.