Annihilators of powers of the maximal ideal in an Artinian Gorenstein ring

Question

Let $(R,\mathfrak{m},k)$ be a commutative Artinian Gorenstein ring. Let $n$ be such that $\mathfrak{m}^n\neq0=\mathfrak{m}^{n+1}$. I see that $\mathfrak{m}^n=(0:\mathfrak{m})$, is it true that $\mathfrak{m}^{n-1}=(0:\mathfrak{m}^2)$? And more generally $\mathfrak{m}^{n-i}=(0:\mathfrak{m}^{i+1})$?