Let $\phi$ be a lower semicontinuous submeasure, that is, a function $\mathcal{P}(\mathbf{N}) \to [0,\infty]$ which is monotone, subadditive, and $$ \phi(A)=\lim_{n\to \infty} \phi(A \cap [1,n]) $$ for all subsets $A$ of positive integers. Moreover, it assumed that $\phi(\emptyset)=0$.
Finally, for each $A\subseteq \mathbf{N}$, set $$ \|A\|_\phi=\lim_{n\to \infty}\phi(A\setminus [1,n]). $$
I am going to ask for a refinement of Question 2 of this MO thread:
Question. Fix a set $A\subseteq \mathbf{N}$ such that $0<\|A\|_\phi\le \phi(A)<\infty$. Is it true that there exists a subset $B\subseteq A$ such that $0<\|B\|_\phi < \|A\|_\phi$ ?