$$s^T\nabla g_j = 0$$
The effect of such a move on the objective function and constraints can be determined only from higher derivatives. In some cases a move in this direction could reduce the objective function without violating the constraints even though the Kuhn-Tucker conditions are met. Therefore, the Kuhn-Tucker conditions are necessary but not sufficient for optimality.
Where $\boldsymbol{s}$ is the feasible direction and $g_j$ are inequality constraints.
Why exactly does this case need to be considered? If I understood it correctly it's the only case that actually leads to the KKT conditions to only be necessary. English is not my first language, maybe it's somewhere in the document but I would like a deeper explanation than
In some cases a move in this direction could reduce the objective function without violating the constraints
I can't really derive it from a plot.