### 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
add_child(efs);
	
	FileDialog::get_icon_func = _file_dialog_get_icon;
	FileDialog::register_func = _file_dialog_register;
	FileDialog::unregister_func = _file_dialog_unregister;
	EditorFileDialog::get_icon_func = _file_dialog_get_icon;
	EditorFileDialog::register_func = _editor_file_dialog_register;
	EditorFileDialog::unregister_func = _editor_file_dialog_unregister;
	editor_export = memnew(EditorExport);
	add_child(editor_export);
	
	EditorColorMap::create();
	EditorTheme::initialize();
	theme = create_custom_theme();
	DisplayServer::set_early_window_clear_color_override(true, theme->get_color(SNAME("background"), EditorStringName(Editor)));
	register_exporters();
	EDITOR_DEF("interface/editor/save_on_focus_loss", false);
	EDITOR_DEF("interface/editor/show_update_spinner", false);
	EDITOR_DEF("interface/editor/update_continuously", false);
	EDITOR_DEF("interface/editor/localize_settings", true);
	EDITOR_DEF_RST("interface/scene_tabs/restore_scenes_on_load", true);
	EDITOR_DEF_RST("interface/inspector/default_property_name_style", EditorPropertyNameProcessor::STYLE_CAPITALIZED);
	EditorSettings::get_singleton()->add_property_hint(PropertyInfo(Variant::INT, "interface/inspector/default_property_name_style", PROPERTY_HINT_ENUM, "Raw,Capitalized,Localized"));
	EDITOR_DEF_RST("interface/inspector/default_float_step", 0.001);
	
	
	EditorSettings::get_singleton()->add_property_hint(PropertyInfo(Variant::FLOAT, "interface/inspector/default_float_step", PROPERTY_HINT_RANGE, "0.0000001,1,0.0000001"));
	EDITOR_DEF_RST("interface/inspector/disable_folding", false);
	EDITOR_DEF_RST("interface/inspector/auto_unfold_foreign_scenes", true);
	EDITOR_DEF("interface/inspector/horizontal_vector2_editing", false);