  We consider a continuous time simple random walk on a subset of the square
lattice with wired boundary conditions: the walk transitions at unit edge rate
on the graph obtained from the lattice closure of the subset by contracting the
boundary into one vertex. We study the cover time of such walk, namely the time
it takes for the walk to visit all vertices in the graph. Taking a sequence of
subsets obtained as scaled lattice versions of a nice planar domain, we show
that the square root of the cover time normalized by the size of the subset, is
tight around $\frac{1}{\sqrt{\pi}} \log N - \frac{1}{4 \sqrt{\pi}} \log \log
N$, where $N$ is the scale parameter. This proves an analog, for the wired
case, of a conjecture by Bramson and Zeitouni from 2009. The proof is based on
comparison with the extremal landscape of the discrete Gaussian free field.
