Logarithmic lower-bound conjecture for random-walk speed-up

Let $G$ be a graph on $n$ vertices, let $k$ satisfy $n\geq k\geq 1$, and write $S^k(G)$ for the speed-up in covering the graph. Logarithmic speed-up conjecture. For any graph $G$ and any $n\geq k\geq 1$,

$$S^k(G)\geq\Omega(\log k).$$

Questions about minimal and maximal speed-up as functions of $k$ remain open; this is the conjectured universal logarithmic lower bound, complementing the proposed linear upper bound.