### 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/editor_build_profile.h"
#include "editor/editor_command_palette.h"
#include "editor/editor_data.h"
#include "editor/editor_feature_profile.h"
#include "editor/editor_folding.h"
#include "editor/editor_help.h"
#include "editor/editor_inspector.h"
#include "editor/editor_interface.h"
#include "editor/editor_layouts_dialog.h"
#include "editor/editor_log.h"
#include "editor/editor_native_shader_source_visualizer.h"
#include "editor/editor_paths.h"
#include "editor/editor_plugin.h"
#include "editor/editor_properties.h"
#include "editor/editor_property_name_processor.h"
#include "editor/editor_quick_open.h"
#include "editor/editor_resource_preview.h"
#include "editor/editor_run.h"
#include "editor/editor_run_native.h"
#include "editor/editor_scale.h"
#include "editor/editor_settings.h"
#include "editor/editor_settings_dialog.h"
#include "editor/editor_themes.h"
#include "editor/editor_translation_parser.h"
#include "editor/editor_undo_redo_manager.h"
#include "editor/export/editor_export.h"
#include "editor/export/export_template_manager.h"
#include "editor/export/project_export.h"
#include "editor/fbx_importer_manager.h"
#include "editor/filesystem_dock.h"