I have an orthogonal polygon (all edges are horizontal or vertical) which is convex (no holes in any row of column of the polygon).

I would like to cover as much as possible of this orthogonal polygon using at most $k$ rectangles.

I know the minimum cover version of this problem has been extensively studied, e.g. https://dl.acm.org/doi/pdf/10.1145/800057.808678

However, I am struggling to find results that extend to the maximum coverage version, i.e., instead of covering the whole polygon with as few rectangles as possible, I want to cover as much of the polygon as possible knowing I can only use $k$ rectangles.

Any leads?