  We present high statistics simulation data for the average time $\langle
T_{\rm cover}(L)\rangle$ that a random walk needs to cover completely a
2-dimensional torus of size $L\times L$. They confirm the mathematical
prediction that $\langle T_{\rm cover}(L)\rangle \sim (L \ln L)^2$ for large
$L$, but the prefactor {\it seems} to deviate significantly from the supposedly
exact result $4/\pi$ derived by A. Dembo {\it et al.}, Ann. Math. {\bf 160},
433 (2004), if the most straightforward extrapolation is used. On the other
hand, we find that this scaling does hold for the time $ T_{\rm N(t)=1}(L)$ at
which the average number of yet unvisited sites is 1, as also predicted
previously. This might suggest (wrongly) that $\langle T_{\rm cover}(L)\rangle$
and $T_{\rm N(t)=1}(L)$ scale differently, although the distribution of
rescaled cover times becomes sharp in the limit $L\to\infty$. But our results
can be reconciled with those of Dembo {\it et al.} by a very slow and {\it
non-monotonic} convergence of $\langle T_{\rm cover}(L)\rangle/(L \ln L)^2$, as
had been indeed proven by Belius {\it et al.} [Prob. Theory \& Related Fields
{\bf 167}, 1 (2014)] for Brownian walks, and was conjectured by them to hold
also for lattice walks.
