This question branches from Taras Banakh's recent question on a cardinal characteristic connected to families of partitions that are directed in the ordering of partition refinement.

A partition $\mathcal P$ of $\omega$ is said to be *finitary* if there is $k\in \omega$ such that for every $P\in {\cal P}$ we have $|P|\leq k$.

A family $\mathfrak P$ of partitions of $\omega$ is called *directed* if for any two partitions $\mathcal A,\mathcal B\in\mathfrak P$ there exists a partition $\mathcal C\in\mathfrak P$ such that each set $S\in\mathcal A\cup\mathcal B$ is contained in some set $C\in\mathcal C$.

Let $\mathfrak P$ is a family of partitions of $\omega$. An infinite subset $D\subset\omega$ is called *$\mathfrak P$-discrete* if for any partition $\mathcal P\in\mathfrak P$ there exists a finite set $F\subset D$ such that for any $P\in\mathcal P$ the intersection $P\cap (D\setminus F)$ contains at most one point.

Let ${\frak P}$ be a directed family of finitary partitions admitting no ${\frak P}$-discrete set. Is there ${\frak C}\subseteq {\frak P}$ with the following properties?

- ${\frak C}$ admits no ${\frak C}$-discrete subset, and
- for all ${\cal C}_1, {\cal C}_2\in {\frak C}$ we have that either ${\cal C}_1$ refines ${\cal C}_2$, or vice versa. (If ${\cal P}, {\cal Q}$ are partitions of $\omega$, we say that ${\cal P}$
*refines*${\cal Q}$ if every member of ${\cal P}$ is contained in some (i.e. exactly one) member of ${\cal Q}$.)