Spiro's bipartite quasikernel improvement conjecture

Let $D$ be a source-free bipartite digraph, meaning that every vertex has a nonempty set of external in-neighbors. A quasikernel of $D$ is an independent set $Q\subseteq V(D)$ such that every vertex $v\in V(D)$ satisfies $\operatorname{dist}_D(Q,v)\le 2$. Spiro's bipartite quasikernel improvement conjecture. There exists $\varepsilon>0$ such that, if $D$ has no directed $2$-cycle and no directed $4$-cycle, then $D$ has a quasikernel $Q$ with

$$|Q|\le \left(\frac12-\varepsilon\right)|V(D)|.$$

Spiro proved a strict improvement over the half bound for the relevant bipartite digraphs apart from the excluded cycle cases. The conjecture asks whether the improvement can be made uniform by a fixed positive $\varepsilon$; it remains open.