This question calls for the necessity of a Noetherian hypothesis in a certain relationship between the properties of the rings concerning the prime ideals. We use the following definitions.
A ring $ R $ is called universally catenary if all $ R $-Algebra of finite type is catenary. (Note that $ R $ must not be noetherian.)
A ring $ R $ is called a ring between if for all integral extension of the ring $ R subseteq S $, each string saturated with prime numbers in $ S $ contracts to a saturated chain of prime numbers in $ R $.
From Ratliff's results we get the following.
Theorem: Any catenary ring Noetherian universally catenary is an intermediate ring.
The proof is quite complicated because it relies on several non trivial results concerning the relations between different conditions of the chain. In particular, it is not clear to me if we can go without noetherianness.
Do we know whether or not we can omit the Noetherian hypothesis in the result above?