Let $M,N$ be (non-zero) finitely generated modules over a regular local ring $(R, \mathfrak m)$ of dimension $d$ such that $M_P, N_P$ are free (non-zero) $R_P$-modules for every prime ideal $P\ne \mathfrak m$ of $R$.
Then is there any elementary way to prove that
$$\text {depth}(M \otimes_R N)=\max \{0, \text {depth}(M)+\text {depth}(N)-d \}\ ?$$
I know that this is essentially contained in Proposition 3.1 of https://doi.org/10.7146/math.scand.a-12871 , but I was wondering if there is a direct proof.
My Thoughts: This is definitely true by Auslander-Litchenbaum if $M\otimes_R N$ is torsion-free. So we only have to deal with the case when $M \otimes_R N$ has non-zero torsion ... unfortunately, in this case, I don't have any idea.
Please help
Thanks