### 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
title_bar = memnew(EditorTitleBar);
	main_vbox->add_child(title_bar);
	left_l_hsplit = memnew(HSplitContainer);
	main_vbox->add_child(left_l_hsplit);
	left_l_hsplit->set_v_size_flags(Control::SIZE_EXPAND_FILL);
	left_l_vsplit = memnew(VSplitContainer);
	left_l_hsplit->add_child(left_l_vsplit);
	dock_slot[DOCK_SLOT_LEFT_UL] = memnew(TabContainer);
	left_l_vsplit->add_child(dock_slot[DOCK_SLOT_LEFT_UL]);
	dock_slot[DOCK_SLOT_LEFT_BL] = memnew(TabContainer);
	left_l_vsplit->add_child(dock_slot[DOCK_SLOT_LEFT_BL]);
	left_r_hsplit = memnew(HSplitContainer);
	left_l_hsplit->add_child(left_r_hsplit);
	left_r_vsplit = memnew(VSplitContainer);
	left_r_hsplit->add_child(left_r_vsplit);
	dock_slot[DOCK_SLOT_LEFT_UR] = memnew(TabContainer);
	left_r_vsplit->add_child(dock_slot[DOCK_SLOT_LEFT_UR]);
	dock_slot[DOCK_SLOT_LEFT_BR] = memnew(TabContainer);
	left_r_vsplit->add_child(dock_slot[DOCK_SLOT_LEFT_BR]);
	main_hsplit = memnew(HSplitContainer);
	left_r_hsplit->add_child(main_hsplit);
	VBoxContainer *center_vb = memnew(VBoxContainer);
	main_hsplit->add_child(center_vb);
	center_vb->set_h_size_flags(Control::SIZE_EXPAND_FILL);
	center_split = memnew(VSplitContainer);
	center_split->set_v_size_flags(Control::SIZE_EXPAND_FILL);
	center_split->set_collapsed(false);
	center_vb->add_child(center_split);
	right_hsplit = memnew(HSplitContainer);
	main_hsplit->add_child(right_hsplit);