We just tried to replicate the issue using the same card (STRIX 1080 Ti) using an ASUS Swift PG27V monitor. The display detects both HDMI ports and DP ports when hot plugging between each in the OS. We encountered no issues. Card detects and outputs using whichever port is used (only have to select either HDMI / DP input in monitor OSD).
Things you can test. Could possibly be a cable problem - try different set of cables. Could also be monitor problem - try another monitor to check. If you still encounter these issues after testing, you can contact local ASUS support center and send the card to be tested and possibly RMA'd. (RMA procedure details here: https://rog.asus.com/forum/showthread.php?4813-ASUS-Worldwide-RMA-Guide
Good luck. Please feel free to update us regarding progress / outcome.