The StarTech.com 4-Port DisplayPort Switch provides you with the option to connect 4 devices to the same monitor. It only has 1 output.
It is necessary a splitter (Hub)
More details about the maximum resolution per video port.
I'm not sure if this is a solution for your problem but you could try setting the DP output as the default primary video port in the BIOS:
Under "Advanced - Devices - Video" the value "IGD primary video port" by default is set to "auto". Set it to "dp", save and reboot.
It did help me with a like problem connecting to a Dell Multi-input monitor, but in my case for HDMI. I expect this to work also for DP, but as said, I have not tried this myself. Good luck.