Fejes Tóth's hexagonal density conjecture for generalized Minkowski arrangements

For $0\leq\mu<1$, let $u,v\in\mathbb{R}^2$ be unit vectors making an angle of $\frac{\pi}{3}$, and let $\mathcal{F}_{\mathrm{hex}}(\mu)$ be the family of disks of radius $1+\mu$ whose centers are the lattice

$$\{ku+mv:k,m\in\mathbb{Z}\}.$$

For an arrangement $\mathcal{F}$, let $\delta(\mathcal{F})$ denote its density and let $\delta_U(\mathcal{F})$ denote its upper density. A $\mu$-arrangement is an arrangement in which no disk overlaps the $\mu$-core of another disk.

Fejes Tóth's hexagonal density conjecture. For any $\mu\in(\sqrt{3}-1,1)$ and any $\mu$-arrangement $\mathcal{F}$ in $\mathbb{R}^2$,

$$\delta(\mathcal{F})\leq\delta_U(\mathcal{F}_{\mathrm{hex}}(\mu)).$$

The conjecture asserts that the upper density of the hexagonal construction is an upper bound for every $\mu$-arrangement in the indicated range. It extends the known optimality of the hexagonal arrangement for $\mu\leq\sqrt{3}-1$ to the regime where the basic hexagonal arrangement no longer covers the plane.