Rigidity conjecture for smooth structures on even-dimensional manifold bundles

Let $M\to B$ and $M'\to B$ be tangentially homeomorphic smooth manifold bundles with common fiber a closed oriented even-dimensional manifold, and let $\Theta(M',M)$ denote their stable smooth structure class. Rigidity conjecture. The stable smooth structure class vanishes:

$$\Theta(M',M)=0.$$

Equivalently, rationally and stably, there are no exotic smooth structures on manifold bundles with closed oriented even-dimensional fibers. This is motivated by the vanishing of the relative higher Igusa–Klein torsion for even-dimensional fibers; the conjecture asserts the corresponding rigidity of the stable smooth structure class.