Entire-function Liouville conjecture for stationary Navier–Stokes fields

Let $v$ be a vector field satisfying the assumptions of the Liouville conjecture for stationary Navier–Stokes fields. In addition, suppose that $v$ is the restriction to $\mathbb R^{3}$ of an entire vector field of exponential type on $\mathbb C^{3}$. Entire-function Liouville conjecture. Then $v$ vanishes identically. This is presented as an apparently easier consequence of the main conjecture, but the source does not provide a resolution.