Exact integer {2}-domination number of three-row grids

Prove that the integer {2}-domination number of the three-row grid graph G_{3,n}=P_3\square P_n satisfies γ^{\{2\}}(G_{3,n})=\lfloor 3n/2\rfloor+1 for every positive integer n.

Background

The paper establishes the upper bound γ^{{2}}(G_{3,n})≤\lfloor 3n/2\rfloor+1 by explicitly constructing integer {2}-dominating functions. It then conjectures that this upper bound is exact for every positive integer n. Although the algorithm introduced later can compute the value for any input n, the paper does not provide a proof of the formula.