Directed discrete filling area conjecture for fine surfaces

Let $C=C_{a,b}$ be a fine cycle whose two oriented lengths are $a$ and $b$, and let $M$ be a regular fine surface with boundary $C$. Its area is the number of fine triangles it contains. Directed discrete FAC, or fine FAC. If $M$ fills $C$ isometrically, then

$$\operatorname{Area}(M)\geq 2ab-a-b.$$

This is the discrete directed counterpart of the directed Finsler filling area conjecture; the source notes that fine fillings exist and that the minimum depends only on $a$ and $b$.