### 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 (difference > 0) {
					String parent = full_path.substr(0, difference);
					int slash_idx = parent.rfind("/");
					slash_idx = parent.rfind("/", slash_idx - 1);
					parent = (slash_idx >= 0 && parent.length() > 1) ? parent.substr(slash_idx + 1) : parent;
					r_filenames.write[set_idx] = parent + r_filenames[set_idx];
				}
			}
			
			bool can_proceed = false;
			RBSet<int>::Element *E = iset.front();
			while (E) {
				String scene_name = r_filenames[E->get()];
				bool duplicate_found = false;
				for (const int &F : iset) {
					if (E->get() == F) {
						continue;
					}
					String other_scene_name = r_filenames[F];
					if (other_scene_name == scene_name) {
						duplicate_found = true;
						break;
					}
				}
				RBSet<int>::Element *to_erase = duplicate_found ? nullptr : E;