I had a very similar issue with a rather expensive LG monitor. The issue was reduced after a firmware update to the monitor, but input auto detection still does not work at all
(I think the current one also has some false advertising, because they claim it supports a 75hz refresh rate, but I've never been able to get more than 60hz out of it. Before the DP 1.4 firmware fix, it was stuck at 30hz sometimes.)