Tmux external clipboard interaction with loadb over SSH

I have the external clipboard option enabled:

$ tmux show-options -g -s set-clipboard

set-clipboard external

And I'm using Kitty terminal with tmux in SSH. When I highlight text with the mouse or use [, the contents get copied to my system clipboard just fine, and I can paste it back into tmux using ]. However, when I pipe as follows:

$ echo -n "Hello World" | tmux loadb -

The contents are correct for ], i.e. when I paste within tmux, but my system clipboard is not updated. I can't figure out why that is, shouldn't it be the same behavior?