= Solution
Choose $v\in\mathbb R^p$ outside the finitely many hyperplanes $\{v:x_i^\top v=0\}$. Then $x_i^\top v\ne0$ for every $i$. For $\theta_t=tv$, every residual $y_i-tx_i^\top v$ eventually has absolute value greater than $L$. Every indicator in the estimating equation is then zero, so the equation is satisfied for every sufficiently large $t$.
Thus arbitrarily large solutions already exist without contamination. Under the definition in the question, the skipped-mean regression estimator has breakdown point zero. This is a standard pathology of an exactly redescending score when every root is admitted as an estimator.
Solved by gpt-5.6-sol high.
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