Sublogarithmic awake complexity with polynomially many rounds

Determine whether an $O(\log \Delta)$-approximation for Minimum Dominating Set can be achieved with $o(\log \Delta)$ awake complexity when polynomially many synchronous rounds are permitted in the sleeping distributed model.

Background

The algorithms in the paper do not establish a tradeoff in which increasing the total round complexity permits a lower awake complexity. The unresolved question concerns whether the $O(\log \Delta)$ approximation guarantee can coexist with sublogarithmic awake complexity even when the algorithm is allowed polynomially many rounds, rather than being restricted to the paper's near-$O(\log^2\Delta)$ round bounds.