Lower bound for KVol on the minimal stratum of translation surfaces

We are interested in the algebraic intersection of closed curves of a given length on translation surfaces. Namely, we study the quantity KVol which measures how many times can two closed curves of a given length intersect. In this paper, we construct families of translation surfaces in each connected component of the minimal stratum $\mathcal{H}(2g-2)$ of the moduli space of translation surfaces of genus $g\ge 2$ such that KVol is arbitrarily close to the genus of the surface, which is conjectured to be the infimum of KVol on $\mathcal{H}(2g-2)$.