Linear-connectivity partition conjecture for highly connected digraphs

Let $D$ be a digraph of order $n$, let $k,t,l obreak\text{ be positive integers}$, and let $\delta(D)$ denote the minimum semidegree of $D$. A digraph is strongly $r$-connected if deleting fewer than $r$ vertices leaves it strongly connected.

Linear-connectivity partition conjecture. There exists a constant $c>0$ such that the vertices of every strongly $c\cdot ktl$-connected digraph $D$ with $\delta(D)\geq n-l$ can be partitioned into $t$ parts $V_1,V_2,\ldots,V_t$ such that each of $D[V_1],\ldots,D[V_t]$ and $D[V_1,\ldots,V_t]$ is a strongly $k$-connected digraph.

The preceding result gives a weaker bound of order $\Omega(k^3l^2t^2\log(2kl))$ on the required connectivity. The conjecture asks whether this can be reduced to a quantity linear in $k$, $t$, and $l$, and is motivated by related partition results for highly connected digraphs.