For an extension of commutative rings $R \subseteq S$ , the extension is said to satisfy
Going Up (GU) property if for every chain of prime ideals $P \subseteq P_0$ of $R$ with $P=Q \cap R$ for some prime ideal $Q$ of $S$, there is a prime ideal $Q_0$ of $S$ such that $Q \subseteq Q_0$ and $P_0=Q_0\cap R$
and is said to satisfy Incomparability (NC) if for any two prime ideals $Q_1,Q_2$ of $S$ , $Q_1 \subseteq Q_2$ and $Q_1\cap R=Q_2\cap R \implies Q_1 =Q_2$ .
Now it is known that , any integral extension satisfies GU and NC property.
My question is : If a ring extension $R \subseteq S$ satisfies GU and NC property, then is it true that the extension is integral ?
If this is not true in general, can some extra condition on $R$ and/or on $S$ force the extension to be integral ?