No keyboard input from emacs

**Describe the bug**
When i start any app with eaf (browser or pdf-viewer), for example, no keyboard input is read.

**To Reproduce**
I install using straight, this is my eaf config:
```emacs-lisp
(use-package eaf
:straight (eaf :type git :host github
:repo "emacs-eaf/emacs-application-framework"
:files ("*.el" "*.py" "core" "app" "*.json")
:includes (eaf-markdown-previewer eaf-browser eaf-pdf-viewer)
:pre-build (("python" "install-eaf.py" "--install" "browser" "markdown-previewer" "pdf-viewer"))))

(use-package eaf-pdf-viewer
:after eaf
:hook
(tex-mode . (lambda ()
(add-to-list 'TeX-view-program-list '("eaf" eaf-pdf-synctex-forward-view))
(add-to-list 'TeX-view-program-selection '(output-pdf "eaf"))))
:config
(eaf-bind-key scroll_up "C-n" eaf-pdf-viewer-keybinding)
(eaf-bind-key scroll_down "C-p" eaf-pdf-viewer-keybinding))

(use-package eaf-markdown-previewer
:after eaf)
```

**Expected behavior**
Emacs keybind working, but no keybind is possible to input.

**Versions (please complete the following info):**
- Archlinux, I3 on VirtualBox:
- GNU Emacs 30.0.50 (build 1, x86_64-pc-linux-gnu, GTK+ Version 3.24.38, cairo version 1.18.0) of 2023-10-09:

**Error logs**
No logging

**Additional context**

manually setting this variable, eaf works as expected:
```emacs-lisp
(setq eaf-wm-name "i3"
eaf-is-member-of-focus-fix-wms t)
```
but something is going wrong 'cause the default `eaf-wm-name` was `""` and `eaf-is-member-of-focus-fix-wms` was `nil`

Start by locating where eaf-wm-name and eaf-is-member-of-focus-fix-wms are defined and where the window-manager detection runs. Reproduce the missing keyboard input with EAF under i3, compare the detected values with the manual settings in the report, and verify that keyboard input works without those settings.

Written by the indexing model from the issue text.