### 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
gui_base->add_child(install_android_build_template);
	remove_android_build_template = memnew(ConfirmationDialog);
	remove_android_build_template->set_text(TTR("The Android build template is already installed in this project and it won't be overwritten.\nRemove the \"res:android/build\" directory manually before attempting this operation again."));
	remove_android_build_template->set_ok_button_text(TTR("Show in File Manager"));
	remove_android_build_template->connect("confirmed", callable_mp(this, &EditorNode::_menu_option).bind(FILE_EXPLORE_ANDROID_BUILD_TEMPLATES));
	gui_base->add_child(remove_android_build_template);
	file_templates = memnew(EditorFileDialog);
	file_templates->set_title(TTR("Import Templates From ZIP File"));
	gui_base->add_child(file_templates);
	file_templates->set_file_mode(EditorFileDialog::FILE_MODE_OPEN_FILE);
	file_templates->set_access(EditorFileDialog::ACCESS_FILESYSTEM);
	file_templates->clear_filters();
	file_templates->add_filter("*.tpz", TTR("Template Package"));
	file = memnew(EditorFileDialog);
	gui_base->add_child(file);
	file->set_current_dir("res:");
	file_export_lib = memnew(EditorFileDialog);
	file_export_lib->set_title(TTR("Export Library"));
	file_export_lib->set_file_mode(EditorFileDialog::FILE_MODE_SAVE_FILE);
	file_export_lib->connect("file_selected", callable_mp(this, &EditorNode::_dialog_action));
	file_export_lib_merge = memnew(CheckBox);
	file_export_lib_merge->set_text(TTR("Merge With Existing"));
	file_export_lib_merge->set_h_size_flags(Control::SIZE_SHRINK_CENTER);
	file_export_lib_merge->set_pressed(true);
	file_export_lib->get_vbox()->add_child(file_export_lib_merge);
	file_export_lib_apply_xforms = memnew(CheckBox);
	file_export_lib_apply_xforms->set_text(TTR("Apply MeshInstance Transforms"));
	file_export_lib_apply_xforms->set_h_size_flags(Control::SIZE_SHRINK_CENTER);
	file_export_lib_apply_xforms->set_pressed(false);
	file_export_lib->get_vbox()->add_child(file_export_lib_apply_xforms);