Local Computation Algorithm for MIS

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Suggested by Mohsen Gaffhari
Source wola19

In the model of Local Computation Algorithms (LCA), given an input graph $G=(V,E)$, an algorithm gets, upon query any vertex $v$ of its choosing, the list of neighbors of $v$. In this model, the current state-of-the-art for the query complexity of computing a Maximal Independent Set (MIS) for graph $G$ of maximum degree at most $\Delta$ is an upper bound of $$ \Delta^{O(\log\log \Delta)} \operatorname{poly}\!\log n $$ queries.

Question: Does there exist a $\operatorname{poly}(\log n, \Delta)$-query LCA for MIS?