### 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
p_layout->set_value(p_section, config_key, names);
		}
		int selected_tab_idx = dock_slot[i]->get_current_tab();
		if (selected_tab_idx >= 0) {
			p_layout->set_value(p_section, "dock_" + itos(i + 1) + "_selected_tab_idx", selected_tab_idx);
		}
	}
	Dictionary floating_docks_dump;
	for (WindowWrapper *wrapper : floating_docks) {
		Control *dock = wrapper->get_wrapped_control();
		Dictionary dock_dump;
		dock_dump["window_rect"] = wrapper->get_window_rect();
		int screen = wrapper->get_window_screen();
		dock_dump["window_screen"] = wrapper->get_window_screen();
		dock_dump["window_screen_rect"] = DisplayServer::get_singleton()->screen_get_usable_rect(screen);
		String name = dock->get_name();
		floating_docks_dump[name] = dock_dump;
		int dock_slot_id = wrapper->get_meta("dock_slot");
		String config_key = "dock_" + itos(dock_slot_id + 1);
		String names = p_layout->get_value(p_section, config_key, "");
		if (names.is_empty()) {
			names = name;
		} else {
			names += "," + name;
		}
		p_layout->set_value(p_section, config_key, names);
	}
	p_layout->set_value(p_section, "dock_floating", floating_docks_dump);
	for (int i = 0; i < vsplits.size(); i++) {
		if (vsplits[i]->is_visible_in_tree()) {