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The only reason I'm nervous is because if the screen is loose, it can only lead to further problems down the road.

I went to Apple today and did that corner test as you requested on a 6+ and it seemed structurally tight as far as I can tell and hear. I didn't get a sense of it clicking like it was loose or something was out of place.
 
To be clear YES, not all 6s have a clicking screen and not all have a loose screen moving around, however they ALL present a very little but noticeable giving of the screen assembly in the metal backplate right next to the volume buttons, seen with a light source, due to there not being support below the screen there, like there is with the rest of the screen assembly. The rest of the screen should be fine and not giving in.

Due to the 6+ being bigger, therefore offering more space for support added this cannot be reproduced on the bigger unit.
 
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