### 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_vb->add_child(dock_float);
	}
	dock_select_popup->reset_size();
	for (int i = 0; i < DOCK_SLOT_MAX; i++) {
		dock_slot[i]->set_custom_minimum_size(Size2(170, 0) * EDSCALE);
		dock_slot[i]->set_v_size_flags(Control::SIZE_EXPAND_FILL);
		dock_slot[i]->set_popup(dock_select_popup);
		dock_slot[i]->connect("pre_popup_pressed", callable_mp(this, &EditorNode::_dock_pre_popup).bind(i));
		dock_slot[i]->set_drag_to_rearrange_enabled(true);
		dock_slot[i]->set_tabs_rearrange_group(1);
		dock_slot[i]->connect("tab_changed", callable_mp(this, &EditorNode::_dock_tab_changed));
		dock_slot[i]->set_use_hidden_tabs_for_min_size(true);
	}
	editor_layout_save_delay_timer = memnew(Timer);
	add_child(editor_layout_save_delay_timer);
	editor_layout_save_delay_timer->set_wait_time(0.5);
	editor_layout_save_delay_timer->set_one_shot(true);
	editor_layout_save_delay_timer->connect("timeout", callable_mp(this, &EditorNode::_save_editor_layout));
	top_split = memnew(VSplitContainer);
	center_split->add_child(top_split);
	top_split->set_v_size_flags(Control::SIZE_EXPAND_FILL);
	top_split->set_collapsed(true);
	VBoxContainer *srt = memnew(VBoxContainer);
	srt->set_v_size_flags(Control::SIZE_EXPAND_FILL);
	srt->add_theme_constant_override("separation", 0);
	top_split->add_child(srt);
	scene_tabs = memnew(EditorSceneTabs);
	srt->add_child(scene_tabs);
	scene_tabs->connect("tab_changed", callable_mp(this, &EditorNode::_set_current_scene));
	scene_tabs->connect("tab_closed", callable_mp(this, &EditorNode::_scene_tab_closed));