### Tested versions

v4.2.stable.official [46dc27791]

### System information

Windows 11- Godot v4.2.stable.official [46dc27791] - Vulkan (Forward+)

### Issue description

When I switch picker mode the next time I open it should be the pick mode I selected last time but it's not

https://github.com/godotengine/godot/assets/46812028/522c8b55-14cf-4781-963f-a465a03c9432



### Steps to reproduce

Toggle a picker mode off and on again

### Minimal reproduction project (MRP)

N/A
#include "main/main.h"
#include "scene/3d/bone_attachment_3d.h"
#include "scene/gui/color_picker.h"
#include "scene/gui/dialogs.h"
#include "scene/gui/file_dialog.h"
#include "scene/gui/link_button.h"
#include "scene/gui/menu_bar.h"
#include "scene/gui/menu_button.h"
#include "scene/gui/panel.h"
#include "scene/gui/panel_container.h"
#include "scene/gui/popup.h"
#include "scene/gui/rich_text_label.h"
#include "scene/gui/split_container.h"
#include "scene/gui/tab_container.h"
#include "scene/main/window.h"
#include "scene/property_utils.h"
#include "scene/resources/image_texture.h"
#include "scene/resources/packed_scene.h"
#include "scene/resources/portable_compressed_texture.h"
#include "scene/theme/theme_db.h"
#include "servers/display_server.h"
#include "servers/navigation_server_3d.h"
#include "servers/physics_server_2d.h"
#include "servers/rendering_server.h"
#include "editor/audio_stream_preview.h"
#include "editor/debugger/editor_debugger_node.h"
#include "editor/debugger/script_editor_debugger.h"
#include "editor/dependency_editor.h"
#include "editor/editor_about.h"
#include "editor/editor_audio_buses.h"