Cohen–Macaulayness from the first Buchsbaum–Rim coefficient

Let $({\bf R},{\mathfrak m})$ be a Noetherian local ring with $\dim {\bf R}\geq 2$, and let $U\subseteq {\mathfrak m}{\bf R}^r$ be a parameter module of ${\bf R}^r$, where $r>0$. Here $\text{br}_1(U)$ denotes the first Buchsbaum–Rim coefficient of $U$, and a ring is unmixed if all its associated primes have the same dimension. Cohen–Macaulayness criterion. The ring ${\bf R}$ is Cohen–Macaulay if and only if ${\bf R}$ is unmixed and

$$\text{br}_1(U)=0.$$

The inequality $\text{br}_1(U)\leq 0$ is known from work of Hayasaka and Hyry; the question is whether equality, together with unmixedness, characterizes Cohen–Macaulay rings.