Conformal concordance implies isotopy of positive conformal classes

Let $M$ be a closed compact manifold admitting a positive-scalar-curvature metric, with $n\geq 5$. Let ${\cal C}^+(M)$ denote the space of positive conformal classes, and let $C_0,C_1\in {\cal C}^+(M)$ be conformal classes. They are conformally concordant if

$$Y(M\times [0,1],M\times\{0,1\};C_0\sqcup C_1)>0.$$

Conformal concordance conjecture. If $C_0$ and $C_1$ are conformally concordant, then they are isotopic in ${\cal C}^+(M)$.

Conformal concordance is already known to be an equivalence relation, and isotopic positive conformal classes are conformally concordant. The conjecture asks whether the converse holds for dimensions $n\geq 5$, paralleling the corresponding question for positive-scalar-curvature metrics.