NPC Pill #5: (P)izza =?= (NP)izza

Recently A. Amarilli (a3nm) posted a question on cs.stackexchange.com about the computational complexity of a Test Round problem of the Google France #Hash Code 2015: the Pizza Regina” problem (March 27th, 2015):

Definition [Pizza Regina problem]

Input: A grid $M$ with some marked squares, a threshold $T\in \mathbb{N}$, a maximal area $A \in\mathbb{N}$

Output: The largest possible total area of a set of disjoint rectangles with integer coordinates in $M$ such that each rectangle includes at least $T$ marked squares and each rectangle has area at most $A$.

The problem can be converted to a decision problem adding a parameter $k \in \mathbb{N}$ and asking:

Question: Does there exist a set of disjoint rectangles satisfying the conditions (each rectangle has integer coordinates in $M$, includes at least $T$ marked squares and has area at most $A$) whose total area is at least $k$ squares?

The problem is clearly in $\mathsf{NP}$, and after struggling a little bit I found that it is $\mathsf{NP}$-hard (so the Pizza Regina problem is $\mathsf{NP}$-complete). This is a sketch of a reduction from MONOTONE CUBIC PLANAR 1-3 SAT: