How to use double tap hyper to show iTerm Hotkey window?

By default, iTerm2 allows to configure Command ⌘; Shift ⇧, Option ⌥ or Control ⌃ as available keys to double tap them to show the hotkey window.

Like many users, I have configured Karabiner to make Caps Lock ⇪ a Hyper key (⌘ + ⇧ + ⌥ + ⌃).

Currently, my hotkey window is bound by iTerm to double tap Command ⌘ to open the hotkey window. Since that sometimes accidentally opens the hotkey window as well as opens the hotkey window when double tapping Hyper (since Hyper includes Command ⌘) I want to remove this redundancy and bind the hotkey window to Hyper.

Since iTerm does not allow me to double tap a self defined combination of hotkeys I want skhd to open the hotkey window.

How can I configure skhd to open iTerms floating hotkey window?

What I want skhd to do on double tapping hyper:

https://user-images.githubusercontent.com/28692167/215226370-48f3ad78-e692-460e-bcbb-a3f36d9689fb.mov

The issue names skhd, Karabiner, and iTerm2 but no repository file or test. Start by checking skhd's hotkey configuration and whether iTerm2 exposes a macOS action for its floating hotkey window; done means documenting a working double-Hyper configuration or confirming that the integration is not available.

Written by the indexing model from the issue text.