Integral closedness of ideals with nearly Gorenstein extended Rees algebras

Let $(A,\mathfrak m)$ be a two-dimensional regular local ring with infinite residue field, and let $I$ be a non-parameter $\mathfrak m$-primary ideal of $A$. The extended Rees algebra $\mathcal{R}'(I)$ is nearly Gorenstein when its canonical ideal contains the sum of the socles of all canonical modules; in the stated setting, this condition implies that $I$ is integrally closed.

Integral-closedness conjecture. If $\mathcal{R}'(I)$ is nearly Gorenstein, then $I$ is integrally closed.

This claim concerns the relationship between the singularity property of an extended Rees algebra and the integral closure of the ideal defining it. The supplied source does not establish whether the assertion is proved or remains open.