In the MWE below, the first and second references link to eq. (I.1), and the third and fourth equation references both link to eq. (2.1). While they should link to eq. (I.1), (1.1), (II.1) and (2.1), respectively.

What causes this error here?

\documentclass[10pt,a4paper,openany]{book}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}

\begin{document}

\numberwithin{equation}{part}
\part{I}
$$\label{eq:I_1} a = b$$

\numberwithin{equation}{chapter}
\chapter{I}
$$\label{eq:chap_1} a = part 1$$

\numberwithin{equation}{part}
\part{II}
$$\label{eq:II_1} a = chap 1$$

\numberwithin{equation}{chapter}
\chapter{2}
$$\label{eq:chap_2} a = chap 2$$

\cref{eq:I_1}, \cref{eq:chap_1}, \cref{eq:II_1}, \cref{eq:chap_2}.

\end{document}