Why can't Safari display this emoji when I am logged in? http://www.halfapp.co uses the  emoji.

In Safari on macOS I see a question mark in a square. See screen.
I also have this issue in emoji used on my Instagram profile.  

Also, in the web inspector, I see the emoji.
And if I use Look Up, the emoji is shown.
Logging in as a guest user I see the emoji.  

No duplicate fonts, no fonts with errors.
Apple Color Emoji is where it should be.  

And the emoji is displaying just fine in this post!

Permissions have been repaired and re-checked, but I can check something specific if that would help.

If I remove "-apple-system" from the CSS font definition the emoji appear (and of course the font changes).

Any ideas?

macOS 10.13, Safari 11.0 (13604.1.38.1.6)
The problem is that auto activation of fonts had been disabled. FontExplorer X Pro was the culprit. 

Safari seems to need auto activation to be able to display emoji correctly.

The solution is to manually re-enable auto activation. In Terminal:

atsutil autoactivation -e

Credit: https://discussions.apple.com/message/32553150#message32553150