Normality conjecture for associated graded rings of prime ideals

Let $R$ be a Cohen--Macaulay ring and let $\mathfrak{p}$ be a prime ideal of finite projective dimension. Write

$$\mathrm G=\bigoplus_{t\geq 0}\mathfrak{p}^t/\mathfrak{p}^{t+1}$$

for the associated graded ring. Assume that $\mathrm G$ is a Cohen--Macaulay domain and that $R/\mathfrak{p}$ is integrally closed. Normality conjecture. If

$$\mathrm G_t=\mathfrak{p}^t/\mathfrak{p}^{t+1}$$

is integrally closed for $t=\dim R/\mathfrak{p}-1$, then $\mathrm G$ is normal. This proposes that integral closedness of one specified graded component, under the stated Cohen--Macaulay and normality hypotheses, forces normality of the entire associated graded ring; the source provides no resolution.