Let $A$ be an integral domain and $R=A[X]$. Let $Q$ be a prime ideal of $A$ and let $P=Q[X]$.
If $R_P$ is integrally closed (in its own fractions field), then is $A_Q$ integrally closed (in its own fraction field)? If not true in general, then does some additional assumption on $A$ make it true (like Noetherian or some restriction on dimension) ?
Is the converse true, i.e. if $A_Q$ is integrally closed, then is $R_P$ integrally closed ?