Kakoune color issue with the tmux split window

I found my new command in kakoune in tmux to split a tmux window will cause the incorrect color output, and I thought it is the tmux split-window issue.

The followings are my experiments:

This is tmux split-window kak (open a new kakoune instance), and the color is correct

And this is tmux split-window kak -c <client> (open to a existing kakoune instance), and the color is incorrect

The incorrect color means doesn’t support true colors, right?
I don’t know whether this issue is caused by tmux or kakoune.

My tmux config, only forcing true colors:

set -g default-terminal "tmux-256color"`
set-option -ga terminal-overrides ",*256col*:Tc"

My kakoune config, only specify colorscheme to palenight:

colorscheme palenight