### 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
right_l_vsplit = memnew(VSplitContainer);
	right_hsplit->add_child(right_l_vsplit);
	dock_slot[DOCK_SLOT_RIGHT_UL] = memnew(TabContainer);
	right_l_vsplit->add_child(dock_slot[DOCK_SLOT_RIGHT_UL]);
	dock_slot[DOCK_SLOT_RIGHT_BL] = memnew(TabContainer);
	right_l_vsplit->add_child(dock_slot[DOCK_SLOT_RIGHT_BL]);
	right_r_vsplit = memnew(VSplitContainer);
	right_hsplit->add_child(right_r_vsplit);
	dock_slot[DOCK_SLOT_RIGHT_UR] = memnew(TabContainer);
	right_r_vsplit->add_child(dock_slot[DOCK_SLOT_RIGHT_UR]);
	dock_slot[DOCK_SLOT_RIGHT_BR] = memnew(TabContainer);
	right_r_vsplit->add_child(dock_slot[DOCK_SLOT_RIGHT_BR]);
	
	vsplits.push_back(left_l_vsplit);
	vsplits.push_back(left_r_vsplit);
	vsplits.push_back(right_l_vsplit);
	vsplits.push_back(right_r_vsplit);
	hsplits.push_back(left_l_hsplit);
	hsplits.push_back(left_r_hsplit);
	hsplits.push_back(main_hsplit);
	hsplits.push_back(right_hsplit);
	for (int i = 0; i < vsplits.size(); i++) {
		vsplits[i]->connect("dragged", callable_mp(this, &EditorNode::_dock_split_dragged));
		hsplits[i]->connect("dragged", callable_mp(this, &EditorNode::_dock_split_dragged));
	}
	dock_select_popup = memnew(PopupPanel);
	gui_base->add_child(dock_select_popup);
	VBoxContainer *dock_vb = memnew(VBoxContainer);
	dock_select_popup->add_child(dock_vb);
	HBoxContainer *dock_hb = memnew(HBoxContainer);