### 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
HashMap<String, int> scene_name_to_set_index;
	for (int i = 0; i < r_filenames.size(); i++) {
		String scene_name = r_filenames[i];
		if (!scene_name_to_set_index.has(scene_name)) {
			index_sets.append(RBSet<int>());
			scene_name_to_set_index.insert(r_filenames[i], index_sets.size() - 1);
		}
		index_sets.write[scene_name_to_set_index[scene_name]].insert(i);
	}
	
	for (int i = 0; i < index_sets.size(); i++) {
		RBSet<int> iset = index_sets[i];
		while (iset.size() > 1) {
			
			for (const int &E : iset) {
				int set_idx = E;
				String scene_name = r_filenames[set_idx];
				String full_path = p_full_paths[set_idx];
				
				scene_name = scene_name.get_basename();
				if (full_path.begins_with("res:")) {
					full_path = full_path.substr(6);
				}
				full_path = full_path.get_basename();
				
				scene_name = scene_name.trim_suffix("/");
				full_path = full_path.trim_suffix("/");
				int scene_name_size = scene_name.size();
				int full_path_size = full_path.size();
				int difference = full_path_size - scene_name_size;