Homotopy invariance of positive-scalar-curvature metric spaces for manifolds with the same normal 2-type

Let $M$ and $N$ be closed $d$-manifolds with $d \geq 5$ and the same normal $2$-type. Assume that both $M$ and $N$ admit metrics of positive scalar curvature. Write $\mathcal{R}^+(M)$ for the space of Riemannian metrics of positive scalar curvature on $M$. Normal 2-type homotopy invariance conjecture. One has a homotopy equivalence

$$\mathcal{R}^+(M) \simeq \mathcal{R}^+(N).$$

This claim extends the paper's preceding special cases for spin and non-spin manifolds to arbitrary closed manifolds with the same normal $2$-type. Its status is not resolved in the supplied text.