Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. It has proven (here) that if $\operatorname{supp}(M) \subseteq \operatorname{supp}(N)$ then $\operatorname{cd}(I,M) \leq \operatorname{cd}(I,N)$. From this fact one guesses that there is a relationship between "$f_I(-)$, the finiteness dimension of modules relative to $I$", and "the support of modules" such as: $$\operatorname{supp}(M) \subseteq \operatorname{supp}(N) \iff f_I(M)\le f_I(N). $$
Can anyone prove this relationship or give a counterexample, please? You can add any assumption that helps, such as "being local", or prove it for equality.
$f_I(M) = \inf\ \{i : H_I^i(M)$ is not finitely generated$\}$ is defined in chapter 9 of the book Local Cohomology. An Algebraic Introduction with Geometric Applications by M. P. BRODMANN and R. Y. SHARP.