Blanket-cover time conjecture for graphs

Let $G=(V,E)$) be a graph. For a random walk on $G$, let $\text{{\bf COV}}[G]$ denote its cover time and let $\text{{\bf BCOV}}[G]$ denote the blanket-cover time, the expected first time at which every vertex $v$ has been visited at least $\pi_v\text{{\bf COV}}[G]$ times, where $\pi_v$ is the stationary probability of $v$. Blanket-cover time conjecture.

$$\text{{\bf BCOV}}[G] = O(\text{{\bf COV}}[G]).$$

This conjecture asserts the equivalence, up to a constant factor, between blanket-cover time and cover time that was stated in the paper introducing blanket time. The source provides no resolution of the conjecture.