The subexponential bound conjecture for disjoint q-kernels

Let $D$ be a finite digraph. A source set is a set of vertices with no in-neighbor outside the set, and $r$-disjoint $q$-kernels are $r$ pairwise disjoint sets that are each independent and reach every vertex by directed paths of length at most $q$.

Subexponential disjoint q-kernel conjecture. There exist constants $\varepsilon>0$ and $r_0$ such that, whenever $r\ge r_0$ and $D$ contains no $(r-1)$-source sets, $D$ contains $r$-disjoint $q$-kernels with

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

The conjecture would improve the paper's known exponential bound, whose growth rate is approximately $2^r$. The source identifies substantial barriers to proving it.