My question is inspired by a question of Dominic van der Zypen. Let $[\omega]^\omega$ denote the set of all infinite subsets of $\omega$.
The reaping number, denoted by $\mathfrak r$, is the minimum cardinality of a set $E\subseteq[\omega]^\omega$ such that the hypergraph $(\omega,E)$ is not $2$-colorable, i.e., every function $f:\omega\to\{0,1\}$ is constant on some element of $E$.
The list reaping number, herein denoted by $\mathfrak l$, is the minimum cardinality of a set $E\subseteq[\omega]^\omega$ such that the hypergraph $(\omega,E)$ is not $2$-list-colorable, i.e., there is a sequence $\langle L_n:n\in\omega\rangle$ of $2$-element sets such that every function $f\in\prod_{n\in\omega}L_n$ is constant on some element of $E$. (I hope the letter $\mathfrak l$ is not already reserved for some other cardinal.)
Plainly $\aleph_1\le\mathfrak l\le\mathfrak r\le\mathfrak c$.
My question: Is $\mathfrak l\lt\mathfrak r$ consistent with ZFC?
My thoughts: I have none. I'm sure the question is too hard for me (though it may be quite easy for experts in set theory) and I have no idea other than to post it here as a question and wait for someone to answer.
