Yamabe product conjecture for $M^{n-1}\times S^1$ and $S^2\times T^2$

Let $(M^{n-1},h)$ be a Yamabe metric with $Y(h)>0$, normalized by $R_h\equiv (n-1)(n-2)$. Let $g(T)=h+dt^2$ on $M^{n-1}\times S^1(T)$, and let $u_{n,T}$ denote the function from the Yamabe ordinary differential equation that depends only on $t$. The Yamabe constant $Y(h+dt^2)$ and the Yamabe invariant $\sigma(M^{n-1}\times S^1)$ are defined by the corresponding conformal and metric infima.

Yamabe product conjecture. The Yamabe constant $Y(g(T))$ is minimized by $u_{n,T}$, and

$$\frac{Y(h+dt^2)}{Y(h_{std}+dt^2)}=\left(\frac{Y(h)}{Y(h_{std})}\right)^{(n-1)/n},$$

which implies

$$\frac{\sigma(M^{n-1}\times S^1)}{\sigma(S^n)}\geq\left(\frac{Y(h)}{Y(h_{std})}\right)^{(n-1)/n}.$$

In particular,

$$\sigma(S^2\times T^2)=\sigma(S^4).$$

The conjecture proposes a quantitative relation between Yamabe constants of a manifold and its product with a circle, and would determine the Yamabe invariant of $S^2\times T^2$. The source gives a partial affirmative result via a theorem of Petean, but does not state that the conjecture has been resolved.