Let R be a noetherian normal domain (if it makes any difference, I'm happy to assume R is also local).

If $p$ is a height one prime, then the localization $R_p$ is a dvr, hence the maximal ideal $pR_p$ is principal, generated by a uniformizer $\pi$.

It's easy to see we can pick $\pi \in p$. Similarly, if $\pi' \in p$ is another uniformizer then there exist $s,t \in R - p$ such that $t\pi' = s\pi$.

Can one do better than this?

More precisely, is there a uniformizer $\pi_0 \in p$ such that any other uniformizer $\pi'$ is of the form $\pi' = s\pi_0$ for $s \in R - p$?