### 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
for (FileDialog *E : file_dialogs) {
		E->invalidate();
	}
	for (EditorFileDialog *E : editor_file_dialogs) {
		E->invalidate();
	}
	_mark_unsaved_scenes();
	
	String export_error;
	Error err = OK;
	if (!export_defer.preset.is_empty() && !EditorFileSystem::get_singleton()->is_scanning()) {
		String preset_name = export_defer.preset;
		
		
		export_defer.preset = "";
		Ref<EditorExportPreset> export_preset;
		for (int i = 0; i < EditorExport::get_singleton()->get_export_preset_count(); ++i) {
			export_preset = EditorExport::get_singleton()->get_export_preset(i);
			if (export_preset->get_name() == preset_name) {
				break;
			}
			export_preset.unref();
		}
		if (export_preset.is_null()) {
			Ref<DirAccess> da = DirAccess::create(DirAccess::ACCESS_RESOURCES);