Directed Finsler filling area conjecture

Let $M$ be a surface with a directed Finsler semimetric $F$, filling without shortcuts a Finsler closed curve $(C,G)$. Let $C^+$ and $C^-$ denote the two orientations of $C$, with directed lengths $a=\operatorname{Len}_G(C^+)$ and $b=\operatorname{Len}_G(C^-)$. Directed Finsler FAC. The un-normalized Holmes--Thompson area satisfies

$$\operatorname{Area}_{\mathrm{uHT}}(M,F)\geq\frac{ab}2.$$

This is the directed analogue of the continuous filling area conjecture; the source gives no general proof or resolution.