Do exotic smooth structures on ${\mathbb R}^4$ vary smoothly?

Question

Is the two-parameter family of exotic smooth structures on ${\mathbb R}^4$ constructed by Taubes and Gompf a smooth family? In other words, is there a submersion $p:V\to\mathbb{R}_+\times\mathbb{R}_+$ of $C^\infty$-manifolds such that $p^{-1}(s,t)=\mathbb{R}^4_{(s,t)}$ for every $(s,t)$? The original paper by Taubes only claims the existence of this two-parameter family of mutually non-isomorphic structures, I couldn't glean the stronger smoothness property from it.