In my book the right headphones also have to be at least open, closed, comfortable and portable, all at the same time. BTW, based on your previous posts I take it you are not that much into headphones, are you? And yet, you are dedicating so much of your time to this site. How come? Anyway, by all means keep it up! Can't imagine what this place would look like without your relentness pursuit of sound (as opposed to pseudo) science.
I see what bigshot is getting at, and yet agree with you on the multiple purposes one may have meaning different equipment for different uses. It is no different whether in headphones or speakers. Monitoring in the field calls for accurate small monitors. You give up some things to keep what is most needed. Mainly no deep bass because they need to be small. Car speakers vs home speakers have different needs. There are no perfect for every purpose transducers. Meaning all involve trade offs. Those trades offs that make sense vary with location and use in mind.
So I have one pretty darn nice set of over the ear phones for use walking about driven by mobile phone or tablet. Another full coverage open phone for non-portable use. Another good enough, sturdy, closed, not terribly expensive set of phones for music recording on the go. In a pinch any could be used for any purposes, and only one is the most balanced, neutral and good sounding of the three I have. I have owned a couple phones better than any I have, but for various reasons did not keep them. I am a bit more uncompromising on my serious home speakers as I can decide the room and other aspects and adapt closer to an optimum level. Even those are a compromise however.
I believe in time with more research and expertise thrown at the issue, headphones will be able to provide the most accurate of possible reproductions. My issue with them over speakers is one of comfort. I haven't had phones that were put them on and feel comfortable for hours and hours good. Some were close enough that I could use them a couple hours when needed. The other thing is really low powerful bass. One can fix that with a subwoofer and phones for non-mobile use.