### 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
#include "editor_node.h"
#include "core/config/project_settings.h"
#include "core/extension/gdextension_manager.h"
#include "core/input/input.h"
#include "core/io/config_file.h"
#include "core/io/file_access.h"
#include "core/io/resource_loader.h"
#include "core/io/resource_saver.h"
#include "core/object/class_db.h"
#include "core/object/message_queue.h"
#include "core/os/keyboard.h"
#include "core/os/os.h"
#include "core/os/time.h"
#include "core/string/print_string.h"
#include "core/string/translation.h"
#include "core/version.h"
#include "editor/editor_string_names.h"