Discrete filling area conjecture for square-celled surfaces

Let $C=C_{2n}$ be a cycle graph of length $2n$, and let $M$ be a square-celled surface with boundary $C$. Distances between vertices are measured in the skeleton graph. Discrete FAC for square-celled surfaces. If $M$ fills $C_{2n}$ isometrically, then it has at least

$$\frac{n(n-1)}2$$

square cells. This formulation is stated to be equivalent to the discrete filling area conjecture for walled surfaces.