### 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
bool has_active_tab = false;
	if (p_config_file->has_section_key(EDITOR_NODE_CONFIG_SECTION, "selected_bottom_panel_item")) {
		int selected_bottom_panel_item_idx = p_config_file->get_value(EDITOR_NODE_CONFIG_SECTION, "selected_bottom_panel_item");
		if (selected_bottom_panel_item_idx >= 0 && selected_bottom_panel_item_idx < bottom_panel_items.size()) {
			
			if (bottom_panel_items[selected_bottom_panel_item_idx].button->is_visible()) {
				_bottom_panel_switch(true, selected_bottom_panel_item_idx);
				has_active_tab = true;
			}
		}
	}
	if (p_config_file->has_section_key(EDITOR_NODE_CONFIG_SECTION, "center_split_offset")) {
		int center_split_offset = p_config_file->get_value(EDITOR_NODE_CONFIG_SECTION, "center_split_offset");
		center_split->set_split_offset(center_split_offset);
		
		if (!has_active_tab) {
			bottom_panel_items[0].control->show(); 
			_bottom_panel_switch(false, 0);
		}
	}
	
	if (p_config_file->has_section_key(EDITOR_NODE_CONFIG_SECTION, "selected_default_debugger_tab_idx")) {
		int selected_default_debugger_tab_idx = p_config_file->get_value(EDITOR_NODE_CONFIG_SECTION, "selected_default_debugger_tab_idx");
		EditorDebuggerNode::get_singleton()->get_default_debugger()->switch_to_debugger(selected_default_debugger_tab_idx);
	}
	
	if (p_config_file->has_section_key(EDITOR_NODE_CONFIG_SECTION, "selected_main_editor_idx")) {