The natural deficiency conjecture for convex drawings of complete graphs

For a drawing $D$ of $K_n$, define its deficiency by

$$\delta(D)=\operatorname{cr}(D)-H(n).$$

The drawing $D$ has the natural deficiency property if, for every vertex $v$ of $K_n$,

$$\delta(D-v)\leq 2\delta(D).$$

Here a convex drawing is a drawing in which all vertices lie on the boundary of the unbounded face. The natural deficiency conjecture. For every $k\geq 2$, every convex drawing of $K_{2k}$ has the natural deficiency property. This is presented as a weakening of the Hill Conjecture and arose in connection with lower bounds for the crossing number of $K_{13}$; the source gives no resolution.