Rosenberg's $S^1$-stability conjecture for the Yamabe invariant

Let $M^n$ be a closed, connected $n$-manifold, let $S^1$ be the circle, and let $\sigma$ denote the Yamabe invariant. Rosenberg's $S^1$-stability conjecture. The Yamabe invariants of $M^n$ and $M^n\times S^1$ satisfy

$$\sigma(M^n)>0\iff \sigma(M^n\times S^1)>0.$$

The conjecture asserts stability of positivity of the Yamabe invariant under taking the product with a circle. The paper's abstract proves the corresponding statement for compact manifolds with non-empty boundary in dimensions $3$, $5$, and $6$, but the closed-manifold conjecture itself is not identified as fully resolved here.