A conjecture on the lengths of filling pairs

A pair $(\alpha ,\beta )$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $\alpha \cup \beta$ in $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In \cite{Aou}, Aougab-Huang conjectured that the length of any filling pair on $M$ is at least $\frac{m_g}{2}$, where $m_g$ is the perimeter of the regular right-angled hyperbolic $(8g-4)$-gon. In this paper, we prove a generalized isoperimetric inequality for disconnected regions and we prove the Aougab-Huang conjecture as a corollary.