Don's reachable-subset length conjecture for finite automata

Let $\mathscr{A}=(Q,\Sigma,\delta)$ be an $n$-state automaton, where $Q$ is its state set and $\delta$ is its transition function. For $S\subseteq Q$ of size $k$, suppose there is a word $w$ such that

$$\delta(Q,w)=S.$$

Don's reachable-subset length conjecture. There is a word $v$ such that $\delta(Q,v)=S$ and whose length is at most

$$|v|\leq n(n-k).$$

The conjecture would make the subset-reachability bound proved for aperiodically $1$-contracting automata universal. The source presents it as a conjecture and gives no resolution, so it remains open.