Potential automorphy of certain non self-dual 3-dimensional Galois representations

In a series of papers, van Geemen and Top have defined a family of surfaces $S_z$ indexed by a nonzero integer parameter $z$, and a compatible family of 3-dimensional Galois representations over $\text{Q}(i)$ attached to each surface. In this note we use recent advancements in potential automorphy and automorphy lifting to show that these compatible families are potentially automophic for all values of $z$, and hence that their L-functions have analytic continuation and a functional equation.