Let $X$ be a smooth projective toric variety over $\mathbb{C}$. It is acted by the compact torus $T=(S^1)^n$.

The $T$-equivariant cohomology $H^*_T(X)$ (with coefficients in a field, say) is an algebra over the ring of the $T$-equivariant cohomology of the point $H^*_T(pt)$. The ideal $H^{>0}_T(pt)\cdot H^*_T(X)$ is clearly two-sided.

Is it true that the quotient algebra $H^*_T(X)/H^{>0}_T(pt)\cdot H^*_T(X)$ is isomorphic to $H^*(X)$ as a graded algebra?