Is the normal fiber cone of any $\mathfrak m$-primary ideal Noetherian?

Question

For an ideal $I$ in a commutative Noetherian ring $R$, let $\overline I$ be the integral closure of $I$.

Now for an ideal $I$ in a Noetherian local ring $(R, \mathfrak m)$ consider the graded ring $\bar {\mathcal F}(I):=\oplus_{n \ge 0} \overline {I^n}/\mathfrak m\overline{I^n}$.

My question is:

If $\sqrt I=\mathfrak m$, then is $\bar {\mathcal F}(I)$ always a Noetherian ring ?

I know this is true if $R$ is analytically unramified, but am not sure what happens otherwise.

Please help.