The conductor class group conjecture for associated graded rings

Let $R$ be a Cohen--Macaulay ring, let $\mathfrak{p}$ be a prime ideal of finite projective dimension, and let $\mathrm G=\bigoplus_{t\geq 0}\mathfrak{p}^t/\mathfrak{p}^{t+1}$ be the associated graded ring. Assume that $\mathrm G$ is not integrally closed, let $\overline{\mathrm G}$ be its integral closure, and let $H$ be the subgroup of $\mathsf{Cl}(\overline{\mathrm G})$ generated by the classes of the prime divisors containing the element used in the localization argument. Free-group conjecture. The group $H$ is free. Examples and general arguments suggest this structure, but the source gives no proof or resolution.