= Solution
The statement is false; the displayed probability is zero. Let $\tau$ be the first exit from the fixed domain $\mathbb D$ and choose $x_n>0$ tending to zero. The Bessel scaling in part b gives
$$
T_{x_n}\overset d=x_n^2T_1.
$$
Hence $T_{x_n}\to0$ in probability, while continuity of the trace gives $\tau>0$ almost surely. Therefore
$$
\mathbb P(T_{x_n}<\tau)\longrightarrow1.
$$
Swallowing $x_n$ before leaving $\mathbb D$ forces a boundary contact away from zero before $\tau$. Taking the limit shows that such a contact occurs almost surely, so
$$
\mathbb P\bigl(\eta([0,\tau])\cap\mathbb R=\{0\}\bigr)=0.
$$
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