Do you know what type of HDMI port the laptop has. It really need hdmi 2.0 to be able to run 1080 at 240.
I could not find your exact laptop on asus site but similar x512 models only have hdmi 1.4.
The older version hdmi 1.4 can only partially do 240 if you set the sample to 4:2:0 rather than the standard 4:4:4.
Not all cards and/or monitors support 4:2:0
Now it could be the rather common issue that the laptop is being a pain and using the setting for the built in display on the external monitor. It is a matter of digging around the display setting to be sure it know you have 2 monitors and they each have the proper setting.