In the early 60s, Harary and Hill conjectured H(n) := 1 4b2 cbn−1 2 cbn−2 2 cbn−3 2 c to be the minimum number of crossings among all drawings of the complete graph Kn. It has recently been shown that this conjecture holds for so-called shellable drawings of Kn. For n ≥ 11 odd, we construct a non-shellable family of drawings of Kn with exactly H(n) crossings. In particular, every edge in our drawings is intersected by at least one other edge. So far only two other families were known to achieve the conjectured minimum of crossings, both of them being shellable.