### 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
if (!distribution_version.is_empty()) {
		info.push_back(distribution_name + " " + distribution_version);
	} else {
		info.push_back(distribution_name);
	}
#ifdef LINUXBSD_ENABLED
	if (!display_server.is_empty()) {
		info.push_back(display_server);
	}
#endif 
	info.push_back(vformat("%s (%s)", driver_name, rendering_method));
	String graphics;
	if (!device_type_string.is_empty()) {
		graphics = device_type_string + " ";
	}
	graphics += rendering_device_name;
	if (video_adapter_driver_info.size() == 2) { 
		String vad_name = video_adapter_driver_info[0];
		String vad_version = video_adapter_driver_info[1]; 
		if (!vad_version.is_empty()) {
			graphics += vformat(" (%s; %s)", vad_name, vad_version);
		} else {
			graphics += vformat(" (%s)", vad_name);
		}
	}
	info.push_back(graphics);
	info.push_back(vformat("%s (%d Threads)", processor_name, processor_count));
	return String(" - ").join(info);
}
Ref<Texture2D> EditorNode::_file_dialog_get_icon(const String &p_path) {