Schoen's Yamabe invariant conjecture for closed 3-manifolds

Let $M$ be a closed $3$-manifold that admits a constant-curvature metric $h$. Here $Y(M)$ denotes the Yamabe invariant of $M$, and $E(h)$ denotes the normalized Einstein-Hilbert energy of $h$. Schoen's conjecture. If $M$ admits a constant-curvature metric $h$, then

$$Y(M)=E(h).$$

This conjecture concerns the relationship between the Yamabe invariant and constant-curvature metrics on closed $3$-manifolds, and is described as one of the most fundamental unsolved problems related to Yamabe invariants.