Gorensteinness of short local rings in terms of the vanishing of Ext and Tor

Let $(R,mathfrak{m})$ be a commutative Noetherian local ring which contains a regular sequence $ underline{x} = x_1,ldots,x_d in mathfrak{m} smallsetminus mathfrak{m}^2 $ such that $ mathfrak{m}^3 subseteq (underline{x}) $. Let $ M $ be a finite $ R $-module with maximal complexity or curvature, e.g., $ M $ can be a nonzero direct summand of some syzygy module of the residue field $ R/mathfrak{m} $. It is shown that the following are equivalent: (1) $R$ is Gorenstein, (2) $mathrm{Ext}_R^{gg 0}(M,R)=0$, and (3) $mathrm{Tor}_{gg 0}^R(M,omega) = 0$, where $omega$ denotes a canonical module of $R$. It gives a partial answer to a question raised by Takahashi. Moreover, the vanishing of $mathrm{Ext}_R^{gg 0}(omega,N)$ for certain $ R $-module $ N $ is also analyzed. Finally, it is studied why Gorensteinness of such local rings is important.