= Solution
Let $g_t$ map out the original filled hull and let $g_t^*$ map out the filled hull generated by $\Phi(\gamma[0,t])$. Both have <hydrodynamic normalization at infinity>. Write
$$
h_t=g_t^*\circ\Phi\circ g_t^{-1},\qquad
W_t=h_t(\xi_t),\qquad a_t=h_t'(\xi_t),\quad b_t=h_t''(\xi_t).
$$
Although the original $\Phi$ is specified only on $N$, this conjugacy is defined in the surviving neighborhood of the tip. Boundary preservation gives a real analytic extension there by the <Schwarz reflection principle>, with $a_t>0$. The identities below are first applied on compactly localized intervals before exit and then continued up to $T$.
Set $u(t)=\operatorname{hcap}(K_t^*)/2$, the capacity clock of the image hull. The <Chordal Loewner equation> for each flow and the chain rule give
$$
\partial_t h_t(z)
=\frac{2\dot u(t)}{h_t(z)-W_t}
-\frac{2h_t'(z)}{z-\xi_t},\qquad W_t=h_t(\xi_t).
$$
Analyticity at the driving point cancels the coefficient of $(z-\xi_t)^{-1}$, forcing $\dot u(t)=a_t^2$. The constant term, computed as in Question 3, is then $\partial_t h_t(\xi_t)=-3b_t$. For a general <SLE> driver $d\xi_t=\sqrt\kappa\,dB_t$, the <Itô formula> consequently gives the <transformed SLE driving function>
$$
dW_t=\sqrt\kappa\,a_t\,dB_t+
\left(\frac\kappa2-3\right)b_t\,dt.
$$
At $\kappa=6$, this becomes
$$
\boxed{dW_t=\sqrt6\,a_t\,dB_t,\qquad
\langle W\rangle_t=6\int_0^t a_s^2ds=6u(t).}
$$
Thus $W$ is a continuous <local martingale> up to the exit time. If the source notation $\xi_t^*$ indexes the image driver by original time, it denotes $W_t$. If it uses the image's own half-plane-capacity time, it denotes $W_{t(u)}$, with $t(u)$ the inverse clock; that time-changed process is also a <local martingale>. This distinguishes the two clocks without changing the requested conclusion.
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