Presentation of a local complete intersection

Question

What is the simplest example of a local (noetherian) complete intersection ring $R$ that can not be presented as $R=S/I$, where $S$ is a regular local ring and $I$ is an ideal generated by a regular sequence?

Note that any definition of a local complete intersection is equivalent to existence of such representation for the completion of $\hat{R}$, but not for $R$ itself.

Answer 1

Such an example can be found in this paper: http://arxiv.org/abs/1109.4921.