Finite-presentation conjecture for the double dimer group

Let $G$ be the group generated by the matrices associated with the three tile-edge types $X$, $Y$, and $Z$, and let $I$ be the $2\times2$ identity matrix. Let $-I$ be the $2\times2$ matrix with $-1$ on the diagonal. Using the boundary-word notation for products of these generators, the conjecture concerns finite presentations of $G$.

Finite-presentation conjecture. $G$ has either of the following two equivalent finite presentations:

$$G = \left\langle X, Y, Z \mathrel{\middle|} \downharpoonleft\hspace{-1.2ex}[{XXX}]\hspace{-1.2ex}\upharpoonright = \downharpoonleft\hspace{-1.2ex}[{yXyX}]\hspace{-1.2ex}\upharpoonright = \downharpoonleft\hspace{-1.2ex}[{yzYX}]\hspace{-1.2ex}\upharpoonright = \downharpoonleft\hspace{-1.2ex}[{YzzX}]\hspace{-1.2ex}\upharpoonright = \downharpoonleft\hspace{-1.2ex}[{yZxYzX}]\hspace{-1.2ex}\upharpoonright = -I \right\rangle$$

or, equivalently,

$$G = \left\langle X, Y, Z \mathrel{\middle|} \begin{matrix} \downharpoonleft\hspace{-1.2ex}[{ZxxYXXX}]\hspace{-1.2ex}\upharpoonright = \downharpoonleft\hspace{-1.2ex}[{yZyZYYzYX}]\hspace{-1.2ex}\upharpoonright = \downharpoonleft\hspace{-1.2ex}[{ZZxZYzXzX}]\hspace{-1.2ex}\upharpoonright = I, \\ \downharpoonleft\hspace{-1.2ex}[{XXX}]\hspace{-1.2ex}\upharpoonright = \downharpoonleft\hspace{-1.2ex}[{yZxYzX}]\hspace{-1.2ex}\upharpoonright = \downharpoonleft\hspace{-1.2ex}[{yXyX}]\hspace{-1.2ex}\upharpoonright = -I \end{matrix} \right\rangle.$$

The presentation is motivated by reducing the remaining open boundary words using the bone and snake relations. The source presents this as the group description in the conjecture and does not state a proof or resolution.