I am currently fighting with ASUS UX510UX and HDMI issues. Before this, I had UX305, which I gave up believing that its HDMI port had broken suddenly down. I was mistaken.
With UX510UX I soon encoutered the same situation, HDMI ceased to function. This happened with the 1607 update and I rolled it back and this restored HDMI functionality. All was fine fore few weeks and then Win10 updated driver files (you cannot prevent this, unfortunately) and I lost the functionality again.
I was in touch with tech support and they provided me with a installer that overwrote all relevant files and HDMI was restored once again... until today when some kind of a driver update once again took place (you hear that "device connected" sound and screen flickers, then you know that you just had unwated driver update). This time the provided package could not restore HDMI functionality and I am currently trying out reinstalling.
Win10 built-in "Reset" is no good, even with "remove all files" option. Nothing changed and all the bloatware just came back. Now I am performing a factory reinstall with tech support's help.
None of this is likely to help you, but you're not alone and it's unlikely that your hardware is at fault.
You might also be interested that these HDMI issues are occuring with other brands too. From the looks of it, this may be related to all motherboards that have certain Intel chipsets and/or GPU (I believe these are all built into the mobile chip these days). Why this may be of interest is that in Microsoft site some users have reported a success overcoming this issue by locating and buying a "better" HDMI cable. I lack the expertice to speculate if the cables have (or lack) features to cause these kinds of issues (let's imagine, if the driver suddenly relies on some signal/data/other from the monitor before activating...), but this is something I will look into, as I move along with all these attempts to restore HDMI to my laptop.
If you discover something, please let others know - and if you have any influence towards the brand or MS, please try to make them care about this issue :)