Ties On The Hull

A search range that is too narrow. Several counts are simultaneously optimal at that $\lambda$, so the search stops short of the slope that isolates $k$. Widen the range to $[-10^{18}, 10^{18}]$, and the reported counts then bracket $k$ so the answer can still be read off.

Collinear points. Several counts are simultaneously optimal at that $\lambda$, so the solver's internal tie-breaking picks which one it reports. Force it to always report the largest achievable count (or always the smallest), and the reported counts then bracket $k$ so the answer can still be read off.

A non-convex objective. Several counts are simultaneously suboptimal at that $\lambda$, so no slope ever selects the exact count you want. Abandon the trick and write the $O(nk)$ table directly, since no tie-breaking rule can recover a count that is not on the hull.

A sign error on the penalty. Several counts are simultaneously optimal at that $\lambda$, so the solver reports them from the wrong side of $k$. Negate $\lambda$ and re-run the search, and the reported counts then bracket $k$ so the answer can still be read off.