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.