The actual slots look a bit rounded to me, so they probably aren't as deep as you think they are. Most nuts are a lot higher than they need to be, so hopefully you should be fine. As Kick says, try it first without any glue to check the height. String tension will hold it in place.

Remember after stringing, give the neck quite some time to settle down under tension, overnight is good. Then in this order:

1) Set up the neck relief
2) Set the saddle heights for clean, non-buzzing playing
(you may want to repeat 1 then 2 a few times until you are happy)
3) Adjust nut height/slot depth
4) Set the intonation

I do like the black pickguard.