Vertex connectivity in the LOCAL model

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Suggested by Sorrachai Yingchareonthawornchai
Source wola19

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 $O(\nu k)$?