On duality in the homology algebra of a Koszul complex

The homology algebra of the Koszul complex $K(x_1,\ldots,x_n;R)$ of a Gorenstein local ring $R$ has Poincaré duality if the ideal $I=(x_1,\ldots,x_n)$ of $R$ is strongly Cohen–Macaulay (i.e., all homology modules of the Koszul complex are Cohen–Macaulay) and under the assumption that $\dim R-\operatorname{grade}I\leq4$ the converse is also true.