From what I understand, there seems to be a VQ parameter that you can attach to the end of the inline iframe and set the value to hd720 or hd1080 as the value. After some research, it seems that YouTube once proposed the βVQβ option, and then removed it, and at the time of writing this letter again! In short, your insert should look something like this:
<iframe src="https://www.youtube.com/embed/VIDEO_ID_HERE?vq=hd1080" frameborder="0" allowfullscreen></iframe>
Here is an article related to this that I found during my research: Found here
I checked this briefly on the page and it seems to be working (for now). Hope this helps!
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