= Solution
Put $U=X/\sigma$. The standardized efficient <score function> for scale is
$$
\Lambda(u)=-1-u\frac{f'(u)}{f(u)}.
$$
An optimal B-robust scale M-estimator solves
$$
\sum_{i=1}^n\psi(X_i/\widehat\sigma)=0,
$$
where the optimal bounded score has the clipped-score form
$$
\psi(u)=\operatorname{clip}\bigl(A\{\Lambda(u)-z\},-b,b\bigr).
$$
The constants $A,z$ enforce <Fisher consistency>, the chosen normalization, and the clipping bound $b$ on the <influence function>.
For the <standard normal distribution>, $f'(u)/f(u)=-u$, hence $\Lambda(u)=u^2-1$. The score therefore simplifies to
$$
\psi(u)=\operatorname{clip}\bigl(A(u^2-1-z),-b,b\bigr),
$$
with $z$ chosen so that $\mathbb E\psi(Z)=0$ for $Z\sim N(0,1)$.
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