I got the mr shield tempered glass screen protector from amazon,because most of the tempered glass screen protectors go around the ear piece and I had bad luck with it de laminating there or cracking in the past. It covers the whole screen but doesn't go edge to edge. I am OK because it makes it case friendly. The one that I have shows some fingerprints but not terrible.
Looks like the one I have, not sure what brand because my gf put it on before surprising me with the phone. In the past I've always asked her if she would put them on for me because she used to work at a cellphone store and could get them on perfectly.
I have the G3 and my Pleson was very nice, even saved the screen when I mistakenly put my phone in the same pocket as the keys and had a heavy box push the keys into it. I re-ordered the same brand and still after four months the Pleson on my g3 is nice and smooth.
The one I have now on my g4 plus that looks like yours has a couple of 'rough' areas. I can't see it with my eye but when I run my fingernail over the screen I can feel that most of it is smooth but there are some areas where the texture changes. So far it hasn't affected touch performance or anything like that but I hope that doesn't mean eventually it'll start flaking or prone to scratching
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