Doyle's conjecture on locally univalent circle packings of the hexagonal triangulation

Let $\mathcal{T}_H$ be the hexagonal triangulation obtained from the lattice

$$H=\{v_{m,n}=m+ne^{\frac{\pi i}{3}}:m,n\in\mathbb{Z}\}$$

by connecting lattice vertices at distance $1$. A locally univalent circle packing is a circle packing of $\mathcal{T}_H$ with locally univalent developing map; the regular hexagonal packings and the Doyle spirals $\mathcal{P}_{r_0,x,y}$, defined for $r_0>0$ and $x,y\in\mathbb{R}_+$ by

$$r_{m,n}=r_0x^my^n,\qquad m,n\in\mathbb{Z},$$

are the explicitly known examples.

Doyle's conjecture. There are no locally univalent circle packings of the hexagonal triangulation $\mathcal{T}_H$ other than regular hexagonal packings and Doyle spirals.

The conjecture asserts rigidity of locally univalent circle packings with hexagonal combinatorics: the known regular and spiral families should exhaust the moduli space. However, for the general cases, the conjecture remains open.