= Solution
Write $A_r=[-r,r]\times(0,1]$ and take $r\geq1$. By the <Brownian representation of half-plane capacity>,
$$
\operatorname{hcap}(A_r)
=\lim_{y\to\infty}y\,
\mathbb E_{iy}[\operatorname{Im}B_\tau].
$$
On hitting $A_r$, the exit height is at most one. Moreover, $A_r$ lies in the half-disc of radius $\sqrt{r^2+1}\leq2r$. The <harmonic measure> of that semicircle as viewed from $iy$ is $O(r/y)$: mapping its exterior to $\mathbb H$ by $z\mapsto z+(2r)^2/z$ reduces the estimate to the <Poisson kernel for the upper half-plane> on an interval of length $O(r)$. Consequently
$$
\mathbb E_{iy}[\operatorname{Im}B_\tau]\leq\frac{Cr}{y}
$$
for large $y$, and $\operatorname{hcap}(A_r)\leq Cr$. This is the <half-plane capacity of a low rectangle>.
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