### 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
dock_slot[DOCK_SLOT_LEFT_UR]->add_child(ImportDock::get_singleton());
	dock_slot[DOCK_SLOT_LEFT_UR]->set_tab_title(dock_slot[DOCK_SLOT_LEFT_UR]->get_tab_idx_from_control(ImportDock::get_singleton()), TTR("Import"));
	
	dock_slot[DOCK_SLOT_LEFT_BR]->add_child(FileSystemDock::get_singleton());
	dock_slot[DOCK_SLOT_LEFT_BR]->set_tab_title(dock_slot[DOCK_SLOT_LEFT_BR]->get_tab_idx_from_control(FileSystemDock::get_singleton()), TTR("FileSystem"));
	
	dock_slot[DOCK_SLOT_RIGHT_UL]->add_child(InspectorDock::get_singleton());
	dock_slot[DOCK_SLOT_RIGHT_UL]->set_tab_title(dock_slot[DOCK_SLOT_RIGHT_UL]->get_tab_idx_from_control(InspectorDock::get_singleton()), TTR("Inspector"));
	
	dock_slot[DOCK_SLOT_RIGHT_UL]->add_child(NodeDock::get_singleton());
	dock_slot[DOCK_SLOT_RIGHT_UL]->set_tab_title(dock_slot[DOCK_SLOT_RIGHT_UL]->get_tab_idx_from_control(NodeDock::get_singleton()), TTR("Node"));
	
	dock_slot[DOCK_SLOT_RIGHT_UL]->add_child(history_dock);
	dock_slot[DOCK_SLOT_RIGHT_UL]->set_tab_title(dock_slot[DOCK_SLOT_RIGHT_UL]->get_tab_idx_from_control(history_dock), TTR("History"));
	
	dock_slot[DOCK_SLOT_LEFT_UL]->hide();
	dock_slot[DOCK_SLOT_LEFT_BL]->hide();
	dock_slot[DOCK_SLOT_RIGHT_BL]->hide();
	dock_slot[DOCK_SLOT_RIGHT_UR]->hide();
	dock_slot[DOCK_SLOT_RIGHT_BR]->hide();
	left_l_vsplit->hide();
	right_r_vsplit->hide();
	
	left_r_hsplit->set_split_offset(270 * EDSCALE);
	main_hsplit->set_split_offset(-270 * EDSCALE);
	
	const String docks_section = "docks";
	default_layout.instantiate();
	
	default_layout->set_value(docks_section, "dock_3", "Scene,Import");