### 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
ResourceFormatImporter::get_singleton()->add_importer(import_3d);
		Ref<ResourceImporterImage> import_image;
		import_image.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_image);
		Ref<ResourceImporterTextureAtlas> import_texture_atlas;
		import_texture_atlas.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_texture_atlas);
		Ref<ResourceImporterDynamicFont> import_font_data_dynamic;
		import_font_data_dynamic.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_font_data_dynamic);
		Ref<ResourceImporterBMFont> import_font_data_bmfont;
		import_font_data_bmfont.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_font_data_bmfont);
		Ref<ResourceImporterImageFont> import_font_data_image;
		import_font_data_image.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_font_data_image);
		Ref<ResourceImporterCSVTranslation> import_csv_translation;
		import_csv_translation.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_csv_translation);
		Ref<ResourceImporterWAV> import_wav;
		import_wav.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_wav);
		Ref<ResourceImporterOBJ> import_obj;
		import_obj.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_obj);
		Ref<ResourceImporterShaderFile> import_shader_file;
		import_shader_file.instantiate();
		ResourceFormatImporter::get_singleton()->add_importer(import_shader_file);
		Ref<ResourceImporterScene> import_scene = memnew(ResourceImporterScene(false, true));
		ResourceFormatImporter::get_singleton()->add_importer(import_scene);