Are complete intersection prime ideals of regular rings regular ideals? Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{p}$ be a prime ideal of $R$ which is a complete intersection, i.e. the minimal number of generators of $\mathfrak{p}$ equals its height $h$. Then by Macaulays theorem there is a system of parameters (or equivalently - a regular sequence) $\{a_{1},\dots, a_{h}\}$ which generates $\mathfrak{p}$.
Is it then also true that $\mathfrak{p}$ can be generated by elements $\{b_{1}, \dots, b_{h}\}$ which can be extended to a regular system of parameters for $R$?
Phrased differently, I am asking whether every complete intersection prime ideal in $R$ is regular (in the sense that $R/ \mathfrak{p}$ is regular).
I am asking this question being interested in the situation where $R = \mathbb{C}\{x_{1},\dots, x_{n}\}$ is the ring of convergent power series.

Take $R=\mathbb{C}[x,y]_{(x,y)}$, and take $\mathfrak{p}=(x^2-y^3)$.  $R/\mathfrak{p}$ is not a regular local ring, since it isn't integrally closed in its field of fractions.
