The associated ring to a Gorenstein Artinian ring

Question

Let $(R,\mathfrak{m},k)$ be a commutative local Noetherian ring. We also assume that $R$ is Gorenstein Artinian. Let $G(R)$ be the associated graded ring $\bigoplus_i\mathfrak{m}^i/\mathfrak{m}^{i+1}$. Is it true that $G(R)$ is also Gorenstein? If yes, do you know how to prove it? If not, do you have a conunterexample?

Answer 1

Consider the ring $R=k[[x,y,z]]/(xy,xz,yz,x^3-y^2,x^3-z^2)$, this is a Gorenstein Artinian ring but its Hilbert series is $1+3t+t^2+t^3$, in particular $G(R)$ cannot be Gorenstein. The problem with Hanno's answer is that even though the end degree of $G(R)$ is one-dimensional, the socle might be bigger. For example in our case the initial form $\bar y\in G(R)$ is a socle element living in degree 1. So the claim I made in my question is false.