= Solution
Let $\xi_t=\sqrt\kappa B_t$ and $0<\kappa<4$. We use these precise <Bessel process> facts, which the question permits: for <dimension> $\delta>2$ a process started at $r>0$ exists for all time, never reaches zero, and has a strictly positive all-time minimum. More quantitatively,
$$
\mathbb P_r\left(\inf_{t\geq0}R_t\leq\varepsilon\right)
=(\varepsilon/r)^{\delta-2},\qquad0<\varepsilon<r.
$$
No assertion about <SLE> simplicity or transience is being assumed.
For a real point $x>0$ before its Loewner lifetime, set $Z_t^x=g_t(x)-\xi_t$. Its equation is
$$
dZ_t^x=\frac2{Z_t^x}dt-\sqrt\kappa\,dB_t.
$$
Thus $Z^x/\sqrt\kappa$ is a <Bessel process> of <dimension> $1+4/\kappa>2$. For each rational $x>0$ it stays positive on every finite interval, almost surely. These countably many events hold together. They cover every real $y>0$ as well: choose rational $0<x<y$ and observe that the noise cancels in the difference,
$$
g_t(y)-g_t(x)=(y-x)\exp\left(-2\int_0^t\frac{ds}{Z_s^yZ_s^x}\right)>0.
$$
Consequently $Z^y$ cannot reach zero while $Z^x$ remains positive. The negative side has the identical argument with the reflected <Brownian motion>.
This excludes <Loewner trace> contact with every real point other than zero, rather than just with each fixed rational point. On any finite interval a positive gap at a real point stays bounded away from zero. Continuous dependence of the <Chordal Loewner equation> then solves it on a complex neighborhood of that point, and the analytic flow extends the map conformally across a real interval there. Its real <derivative> is $\exp(-2\int_0^t(Z_s^x)^{-2}ds)>0$, and its image stays away from the driving point. The <Loewner trace> tip therefore cannot be there. We have proved $\gamma[0,\infty)\cap\mathbb R\subseteq\{0\}$.
Now use <Brownian motion> independent increments to restart at every deterministic rational time $s>0$. The centered future maps have <Loewner driver> $\xi_{s+t}-\xi_s$ and therefore the original <SLE> law, independently of the past. Its <Loewner trace> stays in $\mathbb H\cup\{0\}$ by the preceding argument. The inverse map $g_s^{-1}$ extends continuously to the real <boundary> for a continuous trace-generated <compact H-hull>: the finite <Loewner trace> gives a locally connected <boundary>, and the <Caratheodory boundary extension theorem> applies. Its value at $\xi_s$ is $\gamma(s)$. Thus the future in the original plane can meet $K_s\cup\mathbb R$ only at $\gamma(s)$. All these restart conclusions hold simultaneously for rational $s$.
Suppose $\gamma(u)=\gamma(v)$ with $u<v$. The <Loewner trace> cannot be constant throughout $(u,v)$, since its <compact H-hull> capacity increases strictly. By continuity there is a rational $s\in(u,v)$ with $\gamma(s)\ne\gamma(u)$. But $\gamma(v)=\gamma(u)$ lies in $K_s\cup\mathbb R$, contradicting the restart conclusion at $s$. This also excludes any return to the starting point. Hence \b[the <Loewner trace> is simple and lies in the open <complex upper half-plane> at every positive time].
For transience, first strengthen the <boundary> conclusion to avoidance of the <closure> at each fixed real point. Fix $x>0$ and $0<r<x/4$. If the now-simple curve first enters the closed half-disc of radius $r$ about $x$ at time $\sigma$, join its tip to $x$ by the segment inside that disc. Before $\sigma$ the segment misses the <Loewner trace>. Together with the <Loewner trace> and the real interval from zero to $x$, it bounds a pocket whose <boundary> includes the positive side of the <Loewner trace> and $[0,x/2]$. A <Brownian motion> path from high on the imaginary axis reaching that <boundary> arc must enter the pocket through the segment. The <maximum principle for harmonic functions> bounds its <harmonic measure> by the chance of hitting the entire half-disc.
The <boundary> arc maps under $g_\sigma$ to $[\xi_\sigma,g_\sigma(x/2)]$. Its <harmonic measure> from $iY$, as $Y\to\infty$, is
$$
\frac{g_\sigma(x/2)-\xi_\sigma}{\pi Y}+o(Y^{-1}).
$$
The half-disc maps out by $z\mapsto z+r^2/(z-x)$; its semicircle becomes an interval of length $4r$. Its <harmonic measure> has asymptotic $4r/(\pi Y)$. Comparing and letting $Y\to\infty$ yields
$$
g_\sigma(x/2)-\xi_\sigma\leq4r.
$$
But $m_x=\inf_{t\geq0}(g_t(x/2)-\xi_t)>0$ by the <all-time minimum of a transient Bessel process> fact. The curve cannot enter any such disc with $4r<m_x$. Therefore it stays a strictly positive distance from $x$ for all time. Reflection proves the same for fixed $x<0$. This is the <boundary approach forces a small SLE Bessel gap> argument; the <crosscut> comparison is also described in https://arxiv.org/pdf/math/0106036[the boundary-closure lemma in the primary SLE paper].
Map out the simple initial slit at time one. The starting point zero has two distinct real <prime end> images $b_-,b_+$ under $g_1$, lying on opposite sides of $\xi_1$. Conditionally on the initial slit, the image future is a fresh independent <SLE> translated by $\xi_1$. The just-proved fixed-point <closure> avoidance applies to each of $b_--\xi_1$ and $b_+-\xi_1$. Therefore neither $b_-$ nor $b_+$ lies in the <closure> of the mapped future. If a sequence of future <Loewner trace> points approached zero in the original domain, their images would approach one of these two <prime ends>. This is impossible. Thus
$$
R_1:=\inf_{t\geq1}|\gamma(t)|>0\quad\text{almost surely}.
$$
Finally <Scaling invariance of SLE> gives $R_T:=\inf_{t\geq T}|\gamma(t)|\overset d=\sqrt T R_1$. For every fixed $R>0$,
$$
\mathbb P(R_T\leq R)=\mathbb P(R_1\leq R/\sqrt T)\longrightarrow0.
$$
The events of returning to the closed radius-$R$ disc after time $T$ decrease as $T$ increases. Continuity of probability on decreasing events implies that the probability of returns at arbitrarily late times is zero. Taking the countable intersection over positive integer $R$ proves
$$
\boxed{\gamma\text{ is simple and }|\gamma(t)|\longrightarrow\infty\quad\text{almost surely}.}
$$
This proves actual eventual escape, not merely unboundedness of the growing <compact H-hulls>.
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