Problem 101: Vertex Connectivity in the LOCAL Model

In this question, the input is the underlying graph $G=(V,E)$, as well as parameters $\nu,k$ and vertex $v\in V$. The goal is to output either $\bot$ or a subset $S\subseteq V$, such that
• if $\bot$ is the output, there is no $S$ such that $v\in S$ with $|S| \leq \nu$ and $|N(S)| < k$;
• if the output is a set $S$, then $|N(S)| < k$.
It is known that this problem can be solved with $O(\nu k)$ queries, and either time $O(\nu^{3/2} k)$ (deterministic) or $O(\nu k^2)$ (randomized) [NanongkaiSY-19a,NanongkaiSY-19b,ForsterY-19].
Question: Can one achieve time $O(\nu k)$?