I’ve had a couple of EliteBooks that were picky with external displays, and in most cases the DP port acted dead until I tweaked a few settings. What usually helped was disabling “Hybrid Graphics” in BIOS or forcing the system to initialize the external display first; after a reboot, the monitor finally picked up a signal.
Another thing worth checking is the cable itself — some older DP cables won’t handle higher refresh rates, and the laptop just refuses to output anything. Updating BIOS and resetting the docking/adapter setup also fixed the issue once for me, especially after a major Windows update scrambled display configs.