Algebraic intersection for translation surfaces in the stratum H(2)

The setting is a translation surface $X$ in the stratum H(2) of translation surfaces of genus 2, with one conical point. We study the quantity KVol, defined as the supremum, over all pairs of closed curves, of their algebraic intersection, divided by the product of their length, times the volume of X. We provide an explicit sequence L(n, n) of surfaces such that KVol(L(n, n)) tends to 2 when n goes to infinity, 2 being the conjectured infimum for KVol over H(2).