= Solution
Apply the <derivative criterion for Hölder continuity up to a boundary> to the estimate from part (c). For two points at distance $r$, move each vertically to height at least $r$, join them horizontally there, and move back. The two vertical integrals are bounded by
$$
C\int_0^ry^{-1+\alpha}dy=\frac C\alpha r^\alpha,
$$
and the horizontal integral is at most $Cr\,r^{-1+\alpha}=Cr^\alpha$. Thus $h_T$ extends continuously to the bottom edge and satisfies
$$
|h_T(z)-h_T(w)|\leq C'|z-w|^\alpha
$$
on the half-rectangle. In particular, it is almost surely a <Hölder continuous function> there.
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