Banner video starts out low-res on every loop

There are two things happening with the resolution of my banner video (embedded from YouTube):

1. It starts out terribly low-res (I'm guessing 480p), and I can't find a way to force it to load in high-res from the beginning. The HD only kicks in later. This might normally be tolerable, except...

2. When it loops, it loops back into the low-res video, instead of HD, and repeats the process.

Is there a way to get a banner video to simply work in HD? Is it an issue with YouTube? Even if it lengthens the load time, I'd prefer to have that. I've seen Squarespace-built sites that don't have this problem, like this one.

Any help is appreciated.