Karl, I do not know the answer for your question. However, it is true that, after Kawasaki's result, it has had more papers concerning your question. The following paper, due to Peter Schenzel, is such as a point:
Schenzel, Peter. On birational Macaulayfications and Cohen-Macaulay canonical modules. J. Algebra 275 (2004), no. 2, 751--770.
Perhaps, I have to be more precise.
Background:
Let $(A,\mathfrak{m},\mathbb{K})$ be a local domain. We say that $A$ admits a birational Macaulayfication provided there is an extension ring $A\subseteq B\subseteq\mathbb{Q} (A)$ (where $\mathbb{Q} (A)$ denotes the fraction field of $A$) such that $B$ is a finitely generated Cohen-Macaulay $A$-module.
On the other hand, we say that $A$ is \emph{canonically Cohen-Macaulay} (From now on, CCM for the sake of brevity) if the top module of deficiencies $K_A$ (that is, the canonical module for $A$) is a Cohen-Macaulay module.
In this setup, the main result of Schenzel's paper says that a local domain $A$ which admits a dualizying complex (that is, $A$ must be the homomorphic image of a Gorenstein local ring because of Kawasaki-Sharp's theorem) admits a birational Macaulayfication if and only if $A$ is CCM.