Spiro's disjoint generalized quasikernel conjecture

Let $D$ be a digraph. For an integer $s\ge 1$, call a set $S\subseteq V(D)$ an $s$-source set if

$$N^-(S)=\emptyset\quad\text{and}\quad |S|\le s,$$

where $N^-(S)$ is the set of external in-neighbors of $S$. For an integer $q\ge 1$, a $q$-kernel is an independent set $K\subseteq V(D)$ such that every vertex $v\in V(D)$ satisfies $\operatorname{dist}_D(K,v)\le q$. Spiro's disjoint generalized quasikernel conjecture. There exist $\varepsilon,r_0>0$ such that, whenever $r\ge r_0$ and $D$ has no $(r-1)$-source set, $D$ contains $r$ pairwise disjoint $q$-kernels with

$$q\le (2-\varepsilon)^r.$$

This conjecture seeks an exponential bound on the radius needed for many disjoint generalized quasikernels, improving the previously stated bound $2^{r+1}$. Its general validity remains open.