Let $f : 2^n \to \mathbb{R}$ be a nondecreasing submodular function, where $2^n$ is the powers of $\{1, \dots, n\}$. Here nondecreasing means that $f(S) \le f(T)$ for all $S \subseteq T$. And submodularity means that $f(S) + f(T) \ge f(S \cup T) + f(S \cap T)$ for all $S, T$.
Is it true that $$\sum_{|S| \text{ even}} f(S) \le \sum_{|T| \text{ odd}} f(T)?$$ Here $|S|$ is the cardinality of $S$, as usual.
Assuming the inequality you would like to prove has the sum over odd subsets on the RHS, then you could consider the following counter-example for $n=3$: $f(\emptyset)=0$, $f(\{i\})=1$, $f(\{i, j\})=2$, and $f(\{1,2,3\})=2$. $f$ is non-decreasing and submodular, but the sum over even sets is $6$ and the sum over odd sets is only $5$.
To have the sum over odd sets on the LHS also would not work for $n=2$.
