Asymptotic Bounds for Irredundant Covers of Square Grids by 2 x 2 Cards

We study an irredundant covering of an $m \times m$ square grid by $2 \times 2$ cards, where cards may overlap and every card must lie entirely inside the grid. Every square of the grid must be covered, and every card must contain at least one square that is covered only by itself. Let \(F(m)\) deno