The title says it all - Suppose you are given a noetherian Gorenstein local ring $(A,m,k)$ of finite Krull dimension.
Does there exist a local complete intersection ring $B$ such that $A$ is a homomorphic image of $B$?
In the answer to this question
This paper was given
where there is an example of a Gorenstein ring which is not a quotient of a regular ring, but it is not clear to me if it cannot be expressed as a quotient of a local complete intersection.