  We study the diagonal heat-kernel decay for the four-dimensional
nearest-neighbor random walk (on $\Z^4$) among i.i.d. random conductances that
are positive, bounded from above but can have arbitrarily heavy tails at zero.
It has been known that the quenched return probability $\cmss
P_\omega^{2n}(0,0)$ after $2n$ steps is at most $C(\omega) n^{-2} \log n$, but
the best lower bound till now has been $C(\omega) n^{-2}$. Here we will show
that the $\log n$ term marks a real phenomenon by constructing an environment,
for each sequence $\lambda_n\to\infty$, such that $$ \cmss
P_\omega^{2n}(0,0)\ge C(\omega)\log(n)n^{-2}/\lambda_n, $$ with $C(\omega)>0$
a.s., along a deterministic subsequence of $n$'s. Notably, this holds
simultaneously with a (non-degenerate) quenched invariance principle. As for
the $d\ge5$ cases studied earlier, the source of the anomalous decay is a
trapping phenomenon although the contribution is in this case collected from a
whole range of spatial scales.
