Are the 2013 MBP models even compatible with the Ultrafine 5k? IIRC, models earlier than 2016 can only drive the display in 4k, 60hz, and compatibility cuts off at 2014. https://support.apple.com/en-ca/HT207448
That's correct in the sense that with the 2013 MBP you have to use a ThunderBolt 2 to ThunderBolt 3 adapter, which means that the video signa tops at 4K. I remember reading reports of people saying that there were hints that the computer rendered the video at 5k, scaled it down to 4k for ThunderBolt 2 transportation, and the monitor scaled it back to 5K. I am not sure what process is being done, but things looked not bad at all in Sierra. Much better than on a non-retina screen anyway. It will have to do; I'm not upgrading the computer until one with 32 GB of ram is offered.
Anyhow, your question was if the MBP 2103 are compatible with this screen. Officially, I don't think Apple ever pretended that this setup is supported. But it worked ok in Sierra - with the resoluttion limitation. USB hub, webcam, microphone and speakers work too.
I launched SwitchResX and saw that the LG was still set to use Billions of Colors. Selecting Millions of Colors seems to bring back screen content on the LG.
Edit: corrected my mention that this setup is "compatible" to "it worked but is not officially supported".
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