GSAT incompleteness example
The original paper contains such an example (on page 5):
$$
\begin{gather*}
(1 \lor \lnot 2 \lor 3) \land (1 \lor \lnot 3 \lor 4) \land (1 \lor \lnot 4 \lor \lnot 2) \land (1 \lor 5 \lor 2) \land (1 \lor \lnot 5 \lor 2) \land \\
(\lnot 1 \lor \lnot 6 \lor 7) \land (\lnot 1 \lor \lnot 7 \lor 6) \land \dots \land (\lnot 1 \lor \lnot 98 \lor 99) \land (\lnot 1 \lor \lnot 99 \lor 6)
\end{gather*}
$$
The first five clauses force $1$ to hold. Indeed, suppose that $\lnot 1$. Clauses 4 and 5 force $2$, and so clauses 1 and 3 force $3 \land \lnot 4$, contradicting 2. The entire formula is satisfiable, for example by the all-true assignment.

If the starting assignment is not satisfiable or very close to satisfiable, then the authors claim that the large number of clauses of the form $(\lnot 1 \lor \cdot \lor \cdot)$ will ensure that $1$ remains false throughout the algorithm. See if you can substantiate this claim.