In this note, we demonstrate an instance of bounded-degree graphs of size $n$, for which the total variation mixing time for the random walk is decreased by a factor of $\log n/ \log\log n$ if we multiply the edge-conductances by bounded factors in a certain way.
"Sensitivity of mixing times." Electron. Commun. Probab. 18 1 - 6, 2013. https://doi.org/10.1214/ECP.v18-2765