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Trivially, best possible is $$\begin{pmatrix} 1&1&\dots&&1&2\\ \vdots&&&&2&3\\ \vdots&&&& \vdots & \vdots \\ 1&1&\dots &&n-1 &n\\ 1&1&\dots &n-1 &n&n\\ \vdots&&&& \vdots & \vdots \\ n&n&\dots&&n & n \\ \end{pmatrix}$$ with constant antidiagonals, which has $(n-1)(n^2-1)=O(n^3)$ turning points. Or am I missing something?
(BTW is there a Latex command for rising \ddots?)