The generalized filling radius conjecture

Let $(M^n,g)$ be an orientable closed Riemannian manifold. Suppose that $M$ has $\mathbb T^l$-stabilized scalar curvature $R\geq 1$, meaning that there are positive smooth functions $v_1,v_2,\ldots,v_l$ on $M$ such that $(M\times\mathbb T^l,g+\sum_i v_i^2\mathrm d\theta_i^2)$ has scalar curvature given by the $\mathbb T^l$-invariant extension of $R$. Define the filling radius by

$$r_f(M,g):=\inf\{r>0\mid \mathcal K_*:H_n(M,\mathbb Z)\to H_n(U_r(\mathcal K(M)),\mathbb Z)\text{ is the zero map}\},$$

where $\mathcal K:M\to L^\infty(M)$ is the Kuratowski embedding $p\mapsto \operatorname{dist}_g(p,\cdot)$ and $U_r(\mathcal K(M))$ is the $r$-neighborhood of $\mathcal K(M)$ in $L^\infty(M)$.

Generalized filling radius conjecture. The filling radius of $(M,g)$ is no greater than $C(n)$.

Gromov's reduction relates this conjecture in dimension $n-2$ to the aspherical conjecture in dimension $n$. The assertion is presented as a conjecture in the source, and no resolution is supplied there.