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. by Deborah S. Franzblau and Daniel J. Kleitman in their paper "An algorithm for constructing regions with rectangles: Independence and minimum generating sets for collections of intervals", Proceedings of the sixteenth annual ACM symposium on Theory of computing (STOC '84). Association for Computing Machinery, New York, NY, USA, 167–174, DOI:10.1145/800057.808678 (1984).

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?