### 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_tab_move_left = memnew(Button);
	dock_tab_move_left->set_flat(true);
	dock_tab_move_left->set_focus_mode(Control::FOCUS_NONE);
	dock_tab_move_left->connect("pressed", callable_mp(this, &EditorNode::_dock_move_left));
	dock_hb->add_child(dock_tab_move_left);
	Label *dock_label = memnew(Label);
	dock_label->set_text(TTR("Dock Position"));
	dock_label->set_h_size_flags(Control::SIZE_EXPAND_FILL);
	dock_label->set_horizontal_alignment(HORIZONTAL_ALIGNMENT_CENTER);
	dock_hb->add_child(dock_label);
	dock_tab_move_right = memnew(Button);
	dock_tab_move_right->set_flat(true);
	dock_tab_move_right->set_focus_mode(Control::FOCUS_NONE);
	dock_tab_move_right->connect("pressed", callable_mp(this, &EditorNode::_dock_move_right));
	dock_hb->add_child(dock_tab_move_right);
	dock_vb->add_child(dock_hb);
	dock_select = memnew(Control);
	dock_select->set_custom_minimum_size(Size2(128, 64) * EDSCALE);
	dock_select->connect("gui_input", callable_mp(this, &EditorNode::_dock_select_input));
	dock_select->connect("draw", callable_mp(this, &EditorNode::_dock_select_draw));
	dock_select->connect("mouse_exited", callable_mp(this, &EditorNode::_dock_popup_exit));
	dock_select->set_v_size_flags(Control::SIZE_EXPAND_FILL);
	dock_vb->add_child(dock_select);
	if (!SceneTree::get_singleton()->get_root()->is_embedding_subwindows() && !EDITOR_GET("interface/editor/single_window_mode") && EDITOR_GET("interface/multi_window/enable")) {
		dock_float = memnew(Button);
		dock_float->set_icon(theme->get_icon("MakeFloating", EditorStringName(EditorIcons)));
		dock_float->set_text(TTR("Make Floating"));
		dock_float->set_focus_mode(Control::FOCUS_NONE);
		dock_float->set_h_size_flags(Control::SIZE_SHRINK_CENTER);
		dock_float->connect("pressed", callable_mp(this, &EditorNode::_dock_make_selected_float));