Cyclic covers of rings with rational singularities

We examine some recent work of Phillip Griffith on 'etale covers and fibered products from the point of view of tight closure theory. While it is known that cyclic covers of Gorenstein rings with rational singularities are Cohen-Macaulay, we show this is not true in general in the absence of the Gorenstein hypothesis. Specifically, we show that the canonical cover of a $\mathbb{Q}$-Gorenstein ring with rational singularities need not be Cohen-Macaulay.