Classification conjecture for primitive regular hexagonal tilings of the integer lattice

For $x\in\mathbb{Z}^2$, let $s_x=\{x_1,x_1+1\}\times\{x_2,x_2+1\}\subseteq\mathbb{Z}^2$ and let $\bar{s}_x=[x_1,x_1+1]\times[x_2,x_2+1]\subseteq\mathbb{R}^2$. A tile is a set $T$ of squares $s_x$ such that $I(T)=\bigcup_{s_x\in T}\bar{s}_x$ is a topological disk. A tiling of $\mathbb{Z}^2$ is a collection of tiles in which every square $s_x$ belongs to exactly one tile. A tiling is regular if it has the form $T+L$ for some tile $T$ and lattice $L\subseteq\mathbb{Z}^2$, hexagonal if every tile borders exactly six other tiles, and primitive if every regular refinement is either itself or the tiling by individual squares. Let $\mathcal{B}$ be the family of circles, and for each $C\in\mathcal{B}$ let $T_C+L_C$ be the associated regular tiling by $90^\circ$-symmetric tiles.

Classification conjecture. If $\mathcal{T}$ is a primitive, regular, hexagonal tiling of $\mathbb{Z}^2$ by $90^\circ$-symmetric tiles, then

$$\mathcal{T}=T_C+L_C+v$$

for some $C\in\mathcal{B}$ and some $v\in\mathbb{Z}^2$.

The claim predicts that the circle construction accounts for all primitive regular hexagonal tilings of the integer lattice with the stated rotational symmetry. The supplied text does not establish the classification or indicate whether it has been resolved.