### 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
EditorFileSystemDirectory *efsd = EditorFileSystem::get_singleton()->get_filesystem_path(p_path.get_base_dir());
	if (efsd) {
		String file = p_path.get_file();
		for (int i = 0; i < efsd->get_file_count(); i++) {
			if (efsd->get_file(i) == file) {
				String type = efsd->get_file_type(i);
				if (singleton->icon_type_cache.has(type)) {
					return singleton->icon_type_cache[type];
				} else {
					return singleton->icon_type_cache["Object"];
				}
			}
		}
	}
	return singleton->icon_type_cache["Object"];
}
void EditorNode::_build_icon_type_cache() {