  In the present article, we investigate the following deformation problem. Let
$(R,\mathfrak m)$ be a local (graded local) Noetherian ring with a
(homogeneous) regular element $y \in \mathfrak m$ and assume that $R/yR$ is
quasi-Gorenstein. Then is $R$ quasi-Gorenstein? We give positive answers to
this problem under various assumptions, while we present a counter-example in
general. We emphasize that absence of the Cohen-Macaulay condition requires
some delicate studies.
