  A quasi-kernel of a digraph $D$ is an independent set $Q$ such that every
vertex can reach $Q$ in at most two steps. A 48-year conjecture made by P.L.
Erd\H{o}s and Sz\'ekely, denoted the small QK conjecture, says that every
sink-free digraph contains a quasi-kernel of size at most $n/2$.
  Recently, Spiro posed the large QK conjecture, that every sink-free digraph
contains a quasi-kernel $Q$ such that $|N^-[Q]|\geq n/2$, and showed that it
follows from the small QK conjecture.
  In this paper, we establish that the large QK conjecture implies the small QK
conjecture with a weaker constant. We also show that the large QK conjecture is
equivalent to a sharp version of it, answering affirmatively a question of
Spiro. We formulate variable versions of these conjectures, which are still
open in general.
  Not many digraphs are known to have quasi-kernels of size $(1-\alpha)n$ or
less. We show this for digraphs with bounded dichromatic number, by proving the
stronger statement that every sink-free digraph contains a quasi-kernel of size
at most $(1-1/k)n$, where $k$ is the digraph's kernel-perfect number.
