Discrete filling area conjecture for walled surfaces

Let $(M,W)$ be a walled surface whose boundary is a single closed curve of length $2n$, with area $[?]$ defined as the number of self-crossings of the wallsystem. The boundary metric is the metric obtained by counting crossings with the boundary endpoints of the walls. Discrete FAC for walled surfaces. If $(M,W)$ fills isometrically its boundary, then

$$\operatorname{Area}(M,W)\geq\frac{n(n-1)}2.$$

This is the discrete analogue of the filling area conjecture; the equivalent square-celled formulation appears separately in the source.