On minimal higher genus fillings

In this article, we prove that if $(M,g)$ is a genus $G$ orientable surface with a single boundary component $S^1$, and if $(D,g_0)$ is a disc such that interior points are connected by unique geodesics and $$d_{(D,g_0)}(x,y)\ge d_{(M,g)}(x,y)$$ for all $x,y \in \partial M = \partial D$,then$$(1 + \frac{2 G}{\pi}) $\text{Area}$(M,g) \geq $\text{Area}$(D,g_0).$$