= Solution
Use the <Chordal Loewner equation> $\partial_tg_t=2/(g_t-U_t)$ with $U_t=\sqrt\kappa B_t$. For a fixed $z\in\mathbb H$, let $T_z$ be its <Loewner swallowing time>, and write $Z_t=X_t+iY_t=g_t(z)-U_t$ and $J_t=|g_t'(z)|$ before that time. The <Loewner conformal radius> is $\Upsilon_t=Y_t/J_t$, half of the <conformal radius> of $D_t=\mathbb H\setminus K_t$ at $z$. The <Koebe quarter theorem> bounds the <conformal radius> above by four times the distance to the <boundary>. For the reverse comparison, if $\phi:\mathbb D\to D_t$ maps zero to $z$, apply the <Schwarz lemma> to $\phi^{-1}(z+dw)$ on the <unit disc>, where $d=\operatorname{dist}(z,\partial D_t)$; it gives $d\leq|\phi'(0)|$. Thus
$$
\frac12\operatorname{dist}(z,\partial D_t)\leq\Upsilon_t\leq2\operatorname{dist}(z,\partial D_t).
$$
We use the basic trace theorems that, for $\kappa>8$, the <Loewner trace> is continuous and <Transience of chordal SLE> gives $|\gamma(t)|\to\infty$. These facts make its image relatively closed in $\mathbb H$; they do not assume the space-filling conclusion. The <continuity> and transience statements are available in https://annals.math.princeton.edu/wp-content/uploads/annals-v161-n2-p07.pdf[Rohde and Schramm's basic trace theorems].
Take $\rho=\kappa-8>0$ in the supplied <SLE interior-point martingale>. The <derivative> exponent vanishes, leaving
$$
M_t=\Upsilon_t^{a}S_t^{-b},\qquad a=\frac{\kappa-8}{8}>0,\quad b=\frac{\kappa-8}{\kappa}>0,\quad S_t=\frac{Y_t}{|Z_t|}.
$$
A <nonnegative local martingale> is a <supermartingale>. Applying the <optional stopping theorem> after localization at its first hit of a level $R$ gives the <maximal inequality for a nonnegative supermartingale>
$$
\mathbb P\!\left(\sup_{t<T_z}M_t\geq R\right)\leq\frac{M_0}{R}.
$$
Thus $C=\sup_{t<T_z}M_t<\infty$ almost surely.
Suppose the <Loewner trace> avoids some <open ball> $B(z,r)$ whose <closure> is inside $\mathbb H$. Before $T_z$, the whole <open ball> is in $D_t$: a <connected> <open ball> disjoint from the trace cannot be partly in the unbounded <connected component>. Hence $\Upsilon_t\geq r/2$. The bound on $M$ forces $S_t\geq c>0$ throughout this interval, for a positive random constant $c$.
The <Chordal Loewner equation> and its <derivative> yield
$$
\frac{d}{dt}Y_t^2=-4S_t^2,\qquad \frac{d}{dt}\log\Upsilon_t=-\frac{4Y_t^2}{|Z_t|^4},\qquad \frac{d\log\Upsilon_t}{d\log Y_t}=2S_t^2.
$$
The first identity forces $T_z<\infty$, since otherwise $Y_t^2\leq Y_0^2-4c^2t$ becomes negative. At a finite maximal lifetime, $Y_t\to0$; otherwise the continuous <Loewner driving function> and the <ordinary differential equation> continue past that time. Integrating the last identity, using $S_t\geq c$, gives
$$
\Upsilon_t\leq\Upsilon_0\left(\frac{Y_t}{Y_0}\right)^{2c^2}\longrightarrow0,
$$
contradicting $\Upsilon_t\geq r/2$. Thus every fixed rational ball inside $\mathbb H$ is hit almost surely. A countable intersection makes the trace a <dense subset> almost surely, and its relatively closed image, established by <continuity> and transience, then contains all of $\mathbb H$:
$$
\boxed{\mathbb H\subseteq\gamma[0,\infty)\quad\text{almost surely for }\kappa>8.}
$$
This proves <space-filling SLE above parameter eight>. It rules out unvisited open regions, rather than inferring visits merely from membership in the filled <compact H-hulls>. The choice $\rho=\kappa-8$ does not address the critical value eight.
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