### 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 "editor/gui/editor_file_dialog.h"
#include "editor/gui/editor_run_bar.h"
#include "editor/gui/editor_scene_tabs.h"
#include "editor/gui/editor_title_bar.h"
#include "editor/gui/editor_toaster.h"
#include "editor/history_dock.h"
#include "editor/import/audio_stream_import_settings.h"
#include "editor/import/dynamic_font_import_settings.h"
#include "editor/import/editor_import_collada.h"
#include "editor/import/resource_importer_bitmask.h"
#include "editor/import/resource_importer_bmfont.h"
#include "editor/import/resource_importer_csv_translation.h"
#include "editor/import/resource_importer_dynamic_font.h"
#include "editor/import/resource_importer_image.h"
#include "editor/import/resource_importer_imagefont.h"
#include "editor/import/resource_importer_layered_texture.h"
#include "editor/import/resource_importer_obj.h"
#include "editor/import/resource_importer_shader_file.h"
#include "editor/import/resource_importer_texture.h"
#include "editor/import/resource_importer_texture_atlas.h"
#include "editor/import/resource_importer_wav.h"
#include "editor/import/scene_import_settings.h"
#include "editor/import_dock.h"
#include "editor/inspector_dock.h"
#include "editor/multi_node_edit.h"
#include "editor/node_dock.h"
#include "editor/plugin_config_dialog.h"
#include "editor/plugins/animation_player_editor_plugin.h"
#include "editor/plugins/asset_library_editor_plugin.h"
#include "editor/plugins/canvas_item_editor_plugin.h"