The running lights are on a different circuit than the brake/turn signals so this is not too uncommon of an issue. The first thing I recommend you do is take the ground for both running lights and run a jumper wire up to the main trailer ground as this is the most likely the issue. If the running lights still don't work once this is done that leaves two other things.
I recommend you check to see if the vehicle's trailer connector has power on this circuit. If not then you have a bad fuse... view full answer...
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