Planar graphs of bounded degree: mixing time = cover time?
The answer turns out to be no. A counterexample, kindly suggested to me by Balász Gerencsér, is a $\sqrt{N}$ by $\sqrt{N}$ grid, with a path of length $\sqrt{N}$ glued to it. One can show that


the mixing time of a simple random walk on this graph is $O(N\log N)$ (by combining Cheeger's inequality and the Spielman-Teng [1] bound on the spectral gap)
the cover time of a simple random walk on this graph is $\Omega(N^{3/2})$ (by combining Matthews method with an expression of the hitting time using effective resistance)


[1] Spielman, Daniel A., and Shang-Hua Teng. "Spectral partitioning works: Planar graphs and finite element meshes." Linear Algebra and its Applications 421.2-3 (2007): 284-305.