= Solution
Because $\widetilde F$ solves the differential equation from part (c), the <Itô formula> makes $\widetilde F(V_t)$ a <local martingale> before $\tau$. The defining <improper integral> converges at both endpoints: its integrand is asymptotic to $(-u)^{1-d}$ near zero and to $|u|^{d-3}$ at minus infinity. Since $1<d<2$, both exponents are integrable. Hence
$$
0\leq\widetilde F(v)\leq I:=\int_{-\infty}^0\frac{du}{(-u)^{d-1}(1-u)^{4-2d}}<\infty,
$$
so the stopped local martingale is a bounded martingale.
If $\tau_x<\tau_y$, then $V_t\to0$ and $\widetilde F(V_t)\to0$. If $\tau_y<\tau_x$, then $V_t\to-\infty$ and $\widetilde F(V_t)\to I$. The hitting times cannot coincide because $X_t-Y_t$ stays positive. The <optional sampling theorem for a supermartingale> and <bounded convergence theorem> therefore give
$$
\widetilde F(v)=I\,\mathbb P(\tau_y<\tau_x)=I F(v).
$$
Consequently
$$
F(v)=c\widetilde F(v),
\qquad
c=I^{-1}>0,
$$
which is the <Two-sided SLE boundary swallowing probability>.
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