Is the 4x5 chessboard complex a link complement?

I met this manifold before. It is a normal cover of the orbifold $\mathbb{H}^3/\mathrm{PSL}(2,\mathbb{Z}[\zeta])$ where $\zeta=e^{\pi i/3}$.

I suspect that it actually is $\mathbb{H}^3/\ker\left(\mathrm{PSL}(2,\mathbb{Z}[\zeta])\rightarrow \mathrm{PSL}(2,\mathbb{Z}[\zeta]/I)\right)$ where $I$ is the ideal $\langle 2+2\zeta\rangle$.

For $I=\langle 2+\zeta\rangle$, the manifold is a link complement, see Ian Agol's paper on the Thurston Congruence Link for a drawing.

For $I=\langle 3\rangle$, I could show that it is a link complement, but failed at constructing the link.

Figure 1.27 of http://math.berkeley.edu/~matthias/research/matthias_goerner_thesis_print.pdf shows the link.