Gromov's critical decay conjecture for positive scalar curvature metrics

Let $M$ be an orientable $n$-manifold equipped with a complete Riemannian metric of positive scalar curvature. For a complete Riemannian metric on $M$, say that it has at most $C$-quadratic decay of scalar curvature at infinity if, for some basepoint $p\in M$, there is an $R>0$ such that the scalar curvature $R_M(x)$ satisfies

$$R_M(x)>\frac{C}{r_x^2}$$

for every $x\in M$ with $r_x=d(p,x)\geq R$. Gromov's conjecture. There exists a universal critical constant $C_n>0$ such that: (1) for every $C<C_n$, $M$ admits a complete Riemannian metric of positive scalar curvature with at most $C$-quadratic decay at infinity; and (2) if $M$ admits a complete Riemannian metric of positive scalar curvature with $C$-quadratic decay at infinity for some $C>C_n$, then $M$ admits a complete Riemannian metric with uniformly positive scalar curvature. The conjecture proposes a sharp threshold between decay rates of positive scalar curvature and the existence of uniformly positive scalar curvature metrics on open manifolds; the supplied text does not state whether it is resolved.