Erdős–Hajnal–Soukup partition conjecture for infinite digraphs

Let $G$ be a countably infinite directed graph. For an independent vertex set $K$, call $K$ a quasi-sink if every vertex lies at directed distance at most two into $K$, and call an independent vertex set $Q$ a quasi-kernel if every vertex lies at directed distance at most two out of $Q$.

Erdős–Hajnal–Soukup partition conjecture. There is a partition

$$V(G)=A\cup B$$

such that $G[A]$ contains a quasi-sink and $G[B]$ contains a quasi-kernel.

This is proposed as a genuine infinite analogue of the Chvátal–Lovász quasi-kernel theorem. The supplied text gives no resolution.