Can one add reasonable assumptions that we have $depth\ R \geq depth\ M$ for every $R$-module $M$?

Question

Let $(R,m)$ be a commutative Noetherian local ring which is not CM. Let $M$ be a finite $R$-module.
Here, Hanno shows that one can have any inequality between $depth\ R$ and $depth\ M.$ Still, the 2nd part of question has not answer. After a year I edit the question and ask 2nd part as a separate question:

Can one add reasonable assumptions that we have $depth\ R \geq depth\ M$ for every $R$-module $M$?

Answer 1

Reposting my comment as an answer as you said.

For all reasonable rings, this can not happen is a conjecture of Hochster (though last time I talked to him, he was not so sure he believes it now). The conjecture is, such rings have a finite module which is maximal Cohen-Macaulay. So, if $R$ is not CM, such inequalities can not occur. In particular, if such examples exist, it may not be easy to find.