### 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
return project_export && project_export->is_exporting();
}
void EditorNode::show_accept(const String &p_text, const String &p_title) {
	current_menu_option = -1;
	if (accept) {
		accept->set_ok_button_text(p_title);
		accept->set_text(p_text);
		EditorInterface::get_singleton()->popup_dialog_centered(accept);
	}
}
void EditorNode::show_save_accept(const String &p_text, const String &p_title) {
	current_menu_option = -1;
	if (save_accept) {
		save_accept->set_ok_button_text(p_title);
		save_accept->set_text(p_text);
		EditorInterface::get_singleton()->popup_dialog_centered(save_accept);
	}
}
void EditorNode::show_warning(const String &p_text, const String &p_title) {
	if (warning) {
		warning->set_text(p_text);
		warning->set_title(p_title);
		EditorInterface::get_singleton()->popup_dialog_centered(warning);
	} else {
		WARN_PRINT(p_title + " " + p_text);
	}
}
void EditorNode::_copy_warning(const String &p_str) {