Erdős–Soukup partition conjecture for quasi-kernels and quasi-sinks

Let $G=(V,E)$ be a digraph. An independent set $A\subset V$ is a quasi-kernel if every vertex of $G$ can be reached from some vertex of $A$ by a directed path of length at most $2$; dually, $A$ is a quasi-sink if every vertex can reach some vertex of $A$ by a directed path of length at most $2$. For $W\subset V$, write $G[W]$ for the subgraph induced by $W$.

Erdős–Soukup's conjecture. Every digraph $G=(V,E)$ has a partition $(V_0,V_1)$ such that $G[V_0]$ has a quasi-kernel and $G[V_1]$ has a quasi-sink.

The finite analogue, that every finite digraph has a quasi-kernel, is due to Chvátal and Lovász, whereas the direct assertion fails for infinite digraphs, even for tournaments. The conjecture asks whether a simultaneous quasi-kernel/quasi-sink partition always exists.