Makai's extremal conjecture for $d_{2,1}$

For a convex body $K\subset\mathbb R^{2}$, let $d_{2,1}(K)$ be the non-separable lattice quantity defined in the source. The Makai extremal conjecture.

$$\max\{d_{2,1}(K)\mid K\subset\mathbb R^{2}\text{ is a convex body}\}=\frac{\pi\sqrt{3}}{8}=0.68017\ldots,$$

with equality possible only for ellipses. This is the extremal formulation of the upper-bound conjecture stated immediately earlier in the source, so it is merged here as its fullest restatement. The source does not provide evidence that the conjecture has been resolved.