Interpret concordance of smooth structures in terms of smoothability of a homeomorphism

Determine whether the concordance conclusion in Proposition 2.17 (prop:sliced_concordance) yields any concrete smoothability consequence for a homeomorphism f: X → X. Specifically, Proposition 2.17 asserts that if the Casson–Sullivan invariant cs(f) vanishes, then the smooth structures S and f*(S′) on X are concordant (and sliced concordant when X is non-compact). Ascertain whether such a concordance (or sliced concordance) implies that f is (stably) pseudo-smoothable or otherwise provides a direct interpretation in terms of the smoothability or pseudo-smoothability of f.

Background

In Section 2, the paper relates the Casson–Sullivan invariant cs(f) of a homeomorphism f: X → X to smoothing questions via mapping cylinders. Proposition 2.17 (prop:sliced_concordance) shows that if cs(f)=0 then for smooth structures S and S′ on X the pulled-back structure f*(S′) is concordant to S; moreover, if X is non-compact, they are sliced concordant.

While concordance (or sliced concordance) of smooth structures reflects compatibility at the level of lifting the stable tangent microbundle, it does not directly assert that f itself is isotopic or pseudo-isotopic to a diffeomorphism, i.e., that f is (pseudo-)smoothable. The author notes uncertainty about whether such a concordance statement can be translated into a concrete smoothability statement about f. This leaves open whether the vanishing of cs(f), beyond yielding concordance of smooth structures, implies any (stable) pseudo-smoothability property of f or related smooth-structural consequences for the homeomorphism.