Embedding inverse limits of nearly Markov interval maps as attracting sets of planar diffeomorphisms

In this paper we address the following question due to Marcy Barge: For what f:I → I is it the case that the inverse limit of I with single bonding map f can be embedded in the plane so that the shift homeomorphism ${\displaystyle w\hat{f}}$ extends to a diffeomorphism ([BB, Problem 1.5], [BK, Problem 3])? This question could also be phrased as follows: Given a map f:I → I, find a diffeomorphism ${\displaystyle F:{ℝ}^2\to {ℝ}^2}$ so that F restricted to its full attracting set, ${\displaystyle {\bigcap }_{k\ge 0}{F}^k\left({ℝ}^2\right)}$, is topologically conjugate to ${\displaystyle w\hat{f}:(I,f)\to (I,f)}$. In this situation, we say that the inverse limit space, (I,f), can be embedded as the full attracting set of F.