### 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
singleton->editor_file_dialogs.insert(p_dialog);
}
void EditorNode::_editor_file_dialog_unregister(EditorFileDialog *p_dialog) {
	singleton->editor_file_dialogs.erase(p_dialog);
}
Vector<EditorNodeInitCallback> EditorNode::_init_callbacks;
void EditorNode::_begin_first_scan() {
	
	
	if (!DisplayServer::get_singleton()->window_can_draw()) {
		OS::get_singleton()->benchmark_begin_measure("Editor", "First Scan");
		EditorFileSystem::get_singleton()->scan();
		return;
	}
	if (!waiting_for_first_scan) {
		return;
	}
	requested_first_scan = true;
}
Error EditorNode::export_preset(const String &p_preset, const String &p_path, bool p_debug, bool p_pack_only, bool p_android_build_template) {
	export_defer.preset = p_preset;
	export_defer.path = p_path;
	export_defer.debug = p_debug;
	export_defer.pack_only = p_pack_only;
	export_defer.android_build_template = p_android_build_template;
	cmdline_export_mode = true;
	return OK;
}
bool EditorNode::is_project_exporting() const {