  We study the quantity $\mbox{KVol}$ defined as the supremum, over all pairs
of closed curves, of their algebraic intersection, divided by the product of
their lengths, times the area of the surface. The surfaces we consider live in
the stratum $\mathcal{H}(2)$ of translation surfaces of genus $2$, with one
conical point. We provide an explicit sequence $L(n,n)$ of surfaces such that
$\mbox{KVol}(L(n,n)) \longrightarrow 2$ when $n$ goes to infinity, $2$ being
the conjectured infimum for $\mbox{KVol}$ over $\mathcal{H}(2)$.
